Multiple existence results of solutions for quasilinear elliptic equations with a nonlinearity depending on a parameter

Abstract

We provide existence results of multiple solutions for quasilinear elliptic equations depending on a parameter under the Neumann and Dirichlet boundary condition. Our main result shows the existence of two opposite constant sign solutions and a sign changing solution in the case where we do not impose the subcritical growth condition to the nonlinear term not including derivatives of the solution. The studied equations contain the $p$-Laplacian problems as a special case. Our approach is based on variational methods combining super- and sub-solution and the existence of critical points via descending flow.

1 Introduction and statements of main results

In this paper, we consider the existence of nontrivial multiple solutions for the following quasilinear elliptic equation

$$\begin{aligned} \left\{ \begin{array}{ll} -{\mathrm{div}}\, A(x, \nabla u) =\mu f(x,u)&{\text{ in}} \quad \Omega , \\ Bu=0&{\text{ on}} \quad \partial \Omega , \end{array}\right. \quad {\text{(P)}_\mu } \end{aligned}$$

where $\mu >0$ is a parameter and $\Omega \subset \mathbb R ^N$ is a bounded domain with $C^2$ boundary $\partial \Omega$. Here, $Bu=0$ denotes the Dirichlet or Neumann boundary condition, namely $Bu:=u$ or $Bu:=\partial u/\partial \nu$, respectively, where $\nu$ denotes the outward unit normal vector on $\partial \Omega$. Moreover, $A:\overline{\Omega }\times \mathbb R ^N\rightarrow \mathbb R ^N$ is a map which is strictly monotone in the second variable and satisfies certain regularity conditions (see the following assumption $(A)$).

Throughout this paper, we assume that the map $A$ and the nonlinear term $f$ satisfy the following assumptions $(A)$ and $(f)$, respectively:

$(A) \,A(x,y)=a(x,|y|)y$, where $a(x,t)>0$ for all $(x,t)\in \overline{\Omega }\times (0,+\infty )$, $1<p<\infty$ and

• (i) $A\in C^0(\overline{\Omega }\times \mathbb R ^N, \mathbb R ^N) \cap C^1(\overline{\Omega }\times (\mathbb R ^N\setminus \{0\}), \mathbb R ^N)$;

• (ii) there exists $C_1>0$ such that

  $$\begin{aligned} |D_y\,A(x,y)| \le C_1 |y|^{p-2} \quad {\text{ for} \text{ every}}\ x\in \overline{\Omega }, \ \mathrm{and}\ y\in \mathbb R ^N\setminus \{0\}; \end{aligned}$$

• (iii) there exists $C_0>0$ such that

  $$\begin{aligned} D_y\,A(x,y)\xi \cdot \xi \ge C_0 |y|^{p-2}|\xi |^2 \quad {\text{ for} \text{ every} }\xi \in \overline{\Omega }, \ y\in \mathbb R ^N\setminus \{0\}\ {\text{ and}}\, \xi \in \mathbb R ^N; \end{aligned}$$

• (iv) there exists $C_2>0$ such that

  $$\begin{aligned} |D_x\,A(x,y)| \le C_2(1+ |y|^{p-1}) \quad {\text{ for} \text{ every}}\, x\in \overline{\Omega }, \ y\in \mathbb R ^N\setminus \{0\}; \end{aligned}$$

• (v) there exist $C_3>0$ and $1\ge t_0>0$ such that

  $$\begin{aligned} |D_x\,A(x,y)| \le C_3|y|^{p-1}\,\left( -\log |y|\,\right) \end{aligned}$$

  for every $x\in \overline{\Omega }$, $y\in \mathbb R ^N$ with $0<|y|<t_0$.

$(f)\, f$ is a Carathéodory function on $\Omega \times \mathbb R$ with $f(x,0)=0$ for a.e. $x\in \Omega$ and $f$ is bounded on bounded sets.

In this paper, we say that $u\in W^{1,p}(\Omega )$ (resp. $W_0^{1,p}(\Omega )$) is a (weak) solution of $(P)_\mu$ under the Neumann boundary condition (resp. the Dirichlet boundary condition) if

$$\begin{aligned} \int \limits _{\Omega }A(x, \nabla u) \nabla \varphi \,\mathrm{d}x =\mu \int \limits _{\Omega }f(x,u) \varphi \,\mathrm{d}x \end{aligned}$$

for all $\varphi \in W^{1,p}(\Omega )$ (resp. $W_0^{1,p}(\Omega )$) provided the integral on the right-hand side exists. We say that $u$ is a positive (resp. negative) solution of $(P)_\mu$ if $u\in W^{1,p}(\Omega )$ is a solution in the above sense and $u(x)>0$ (resp. $u(x)<0$) for a.e. $x\in \Omega$.

A similar hypothesis to $(A)$ is considered in the study of quasilinear elliptic problems (cf. [28, Example 2.2.] and see [14, 26, 27, 32] too). We also refer to [19, 29, 31] for the generalized $p$-Laplace operators. In particular, for $A(x,y)=|y|^{p-2}y$, that is, ${\mathrm{div}} A(x,\nabla u)$ stands for the usual $p$-Laplacian $\Delta _p u$, we can take $C_0=C_1=p-1$ in $(A)$. Conversely, in the case where $C_0=C_1=p-1$ holds in $(A)$, by the inequalities in Remark 6 (ii) and (iii) in Sect. 2, we see that $a(x,t)=|t|^{p-2}$ whence $A(x,y)=|y|^{p-2}y$. Hence, our equation ${\text{(P)}_\mu }$ contains the corresponding $p$-Laplacian problem as a special case.

The main purpose of this paper is to show the existence of at least three nontrivial solutions for $(P)_\mu$, provided $\mu$ is sufficiently large, without assuming the subcritical growth condition for the term $f$. This is achieved through a variational approach that encompasses both the Dirichlet and Neumann problems. It is well known that for the Dirichlet problems, due to the Poincaré inequality, we can construct a coercive functional corresponding to the equation simply by the truncation of the nonlinearity. This argument does not work for the Neumann problems because for them the Poincaré inequality does not hold. However, in [26], the present authors overcome this difficulty by introducing new functionals (see also Sects. 4, 5) by means of which we can prove under the Neumann boundary condition, an existence result as in the Dirichlet case via super- and sub-solution, that is, the existence of a solution within the ordered interval determined by a sub-solution and a super-solution.

Here, for such a coercive functional under several hypotheses, we show that it has at least three critical points via the descending flow argument which is done in our main abstract result stated as Theorem 11. This result is developed from the one in [5] which deals with the $p$-Laplace operator in place of our generalized operator and a super-linear nonlinearity under the Dirichlet boundary condition (so, a functional as in [5] is not coercive). Furthermore, we point out that we drop the hypothesis regarding $N$ and $p$ imposed in Bartsch and Liu [5] and removed in [7] in the case of Dirichlet boundary condition. We overcome this difficulty by a different way from the one in [7], roughly speaking by constructing a suitable descending flow on $C^1_0(\Omega )$ and $C^1(\overline{\Omega })$ in place of the Sobolev space $W^{1,p}_0(\Omega )$. As a result, although the result in [7] covers only the case of $f(x,u)=o(|u|^{p-1})$ as $|u|\rightarrow 0$, our abstract theorem can provide the result to the cases of $f(x,u)=O(|u|^{p-1})$ as $|u|\rightarrow 0$ (see Sect. 1.1) and concave near zero (see Sect. 1.2) in Neumann problem too.

Moreover, even without assuming the subcritical growth condition, in [11] (for the Dirichlet problems with the $p$-Laplacian) and in [26] (for the Neumann problems with a general operator in the principal part), the existence of multiple solutions can be established. However, the statements in [11] and [26] did not depend on parameters and the nonlinearities treated there were essentially different from those considered here because now we focus on a nonlinearity $f(x,u)$ without the local sign-condition and whose growth condition near $u = 0$ matters only when the parameter $\mu$ is sufficiently large. In addition, since we do not impose the subcritical growth condition, we can handle nonlinearities $f(x,u)$ containing terms like $|u|^{q-2}u$ ($1<q<\infty$) and $e^u$. Based on Theorem 11, we are able to establish the existence of multiple solutions with complete sign information for both Dirichlet and Neumann nonlinear elliptic equations whose principal part is much more general than the $p$-Laplacian.

Let us recall some relevant results for the $p$-Laplacian problems under the Dirichlet boundary condition. Consider

$$\begin{aligned} -\Delta _p u=f(x,u,\mu ) \quad {\text{ in}}\ \Omega , \quad u=0 \quad {\text{ on}}\ \partial \Omega , \end{aligned}$$

(1)

where $\Delta _p u=\mathrm{div}\,(|\nabla u|^{p-2}\nabla u)$ and $\mu$ is a parameter. When the parameter $\mu$ is large, there are few results of multiple existences containing a sign-changing solution (cf. [6, 12]). In [6], Bartsch and Liu treated the nonlinearity $f(x,u,\mu )=\mu f(x,u)$ satisfying $f(x,u)=o(|u|^{p-1})$ as $|u|\rightarrow \infty$ under additional hypotheses. In [12], Carl and the first author consider the nonlinearity $f$ such that $f(x,u,\mu )=\mu |u|^{p-2}u -g(x,u)$ with $g(x,u)=o(|u|^{p-1})$ as $u\rightarrow 0$ and $g(x,u)/|u|^{p-2}u \rightarrow +\infty$ as $|u|\rightarrow \infty$. In this paper, we give a general result which admits $f(x,u)$ to behave like $m(x)|u|^{p-2}u$ near $u=0$ with a bounded sign-changing function $m$ [see Corollary 2 (ii)].

We also mention that many authors studied a positive (or a nonnegative) solution of Eq. (1) (cf. [1, 10, 17, 20, 21, 30]). This occurred, in particular, when the nonlinearity $f$ is concave–convex, that is,

$$\begin{aligned} f(x,u,\mu )=\mu |u|^{q-2}u+|u|^{r-2}u \end{aligned}$$

with $1<q<p<r\le p^*$. In the semilinear case ($p=2$), the study of existence or nonexistence of a positive solution is well known from Ambrosetti et al. [2]. Later, it has been developed by many authors for the $p$-Laplace problem with concave–convex nonlinearity (cf. [4, 18, 20]). One of our purposes is to provide a sign-changing solution when a positive solution exists (see Corollary 4 and Example 1).

For the generalized operator under the Dirichlet boundary condition, we can see the existence of a nontrivial or a positive radially symmetric solution in [29] or [19], respectively. For the Neumann problems with $p$-Laplacian, we refer to [8, 9]. However, there are no results regarding a sign-changing solution. In the present paper, we prove the existence of three solutions of problem $(P)_\mu$: one positive, one negative and one changing its sign, provided the parameter $\mu$ is sufficiently large. We emphasize that in problem $(P)_\mu$ the operator ${\mathrm{div}}A(x,\nabla u)$ is much more general than the $p$-Laplacian, in particular it is not required to be $(p-1)$-homogeneous. Our main result in this direction is Theorem 1 that applies to both Dirichlet and Neumann boundary value problems. Various corollaries of it provide verifiable conditions for the nonlinearity $f(x,u)$ in order to guarantee the conclusion of Theorem 1. Our approach relies on the analysis with respect to the positive and negative cones of a descending flow related to problem $(P)_\mu$.

The contents of the paper:

Section 1.1 contains the statements of our main results without assuming the (local) sign-condition for $f$ and the growth condition for $f$ near zero [see $(H3)$ and $(H6)$].

In Sect. 1.2, we present existence results in the case where the nonlinearity $f$ contains a concave term near zero [see $(\widetilde{H3})$].

Section 2 is devoted to the properties of the general map $A$ in problem $(P)_\mu$.

In Sect. 3, we prove the main abstract theorem (Theorem 11). In Sect. 4, we give the proofs of our results in the case where the parameter is sufficiently large. In Sect. 5, we prove our results in special cases where the parameter $\mu$ is arbitrary.

1.1 Statements of main results

To simplify the notation, we introduce the following spaces:

$$\begin{aligned} W_B&:= W^{1,p}(\Omega ) \quad {\text{ or}} \quad W_B:=W_0^{1,p}(\Omega ), \\ X_B&:= C^1(\overline{\Omega }) \quad {\text{ or}} \quad X_B:=C^1_0(\overline{\Omega }), \\ C^{1,\alpha }_B(\overline{\Omega })&:= C^{1,\alpha }(\overline{\Omega }) \quad {\text{ or}} \quad C^{1,\alpha }_B(\overline{\Omega }):=C^{1,\alpha }_0(\overline{\Omega }) \quad {\text{ for}}\ \alpha \in (0,1) \end{aligned}$$

in the case of $Bu:=\partial u/\partial \nu$ or $Bu:=u$, respectively. Denote the positive cone $P_B:=\{u\in X_B\,;\,u(x)\ge 0\ {\text{ for} \text{ every}}\ x\in \Omega \,\}$ and the closure of $P_B$ in $W_B$ by $\widetilde{P}_B$. For simplicity, we denote the positive cone in $C^1(\overline{\Omega })$ by $P$, and so

$$\begin{aligned} {\mathrm{int}}\,P:=\{u\in C^1(\overline{\Omega })\,;\,u(x)> 0\ {\text{ for} \text{ every}}\ x\in \overline{\Omega }\}. \end{aligned}$$

In this paper, we set $t_\pm :=\max \{\pm t, 0\}$ and so $u_+$ and $u_-$ denote the positive and the negative part of a function $u$, respectively (that is, $u=u_+-u_-$).

Next, we formulate the following hypothesis: there exists $\mu _0\ge 0$ such that

• (H1) for each $\mu > \mu _0$, there exist a super-solution $u_\mu \in W^{1,p}(\Omega )\cap L^\infty (\Omega )_+\setminus \{0\}$ and a sub-solution $v_\mu \in W^{1,p}(\Omega )\cap (-L^\infty (\Omega )_+)\setminus \{0\}$ of ${({P})}_\mu$; Here, we say that $u\in W^{1,p}(\Omega ) \cap L^\infty (\Omega )$ is a super-solution (resp. sub-solution) of $(P)_\mu$ if $u$ satisfies

  $$\begin{aligned} \int \limits _{\Omega }A(x,\nabla u)\nabla \varphi \,\mathrm{d}x&\ge \mu \int \limits _{\Omega }f(x,u)\varphi \,\mathrm{d}x\\ \left(\,{\text{ resp}}.\, \int \limits _{\Omega }A(x,\nabla u)\nabla \varphi \,\mathrm{d}x \right.&\le \left. \mu \int \limits _{\Omega }f(x,u)\varphi \,\mathrm{d}x \,\right)\ \end{aligned}$$

  for every $\varphi \in W_B$ with $\varphi \ge 0$. In addition, in the case of $W_B=W^{1,p}_0(\Omega )$ (that is, Dirichlet boundary problem), we impose $u\ge 0$ resp. $u\le 0$) on $\partial \Omega$ in the sense of trace operator. We introduce several conditions for $f$ which are not necessarily simultaneously assumed in our results.

• (H2) there exist $D_1>0$ and $\delta _0>0$ such that $f(x,u)u\ge -D_1|u|^p$ for every $|u|<\delta _0$ and a.e. $x\in \Omega$;

• (H3) there exist open subsets $\Omega _1$, $\Omega _2$ of $\Omega$, positive constants $\delta _0$, $d_1$ and $d_2$ such that

  $$\begin{aligned} \inf _{x\in \Omega _1} u_\mu (x)\ge d_1, \qquad \sup _{x\in \Omega _2} v_\mu (x)\le -d_2 \quad {\text{ for} \text{ every}}\ \mu \ge \mu _0,\\ f(x,u)>0 \quad {\text{ for} \text{ every}}\ 0<u<\delta _0, \ {\text{ a.e.}}\ x\in \Omega _1,\\ f(x,u)<0 \quad {\text{ for} \text{ every}}\ 0>u>-\delta _0, \ {\text{ a.e.}}\ x\in \Omega _2, \end{aligned}$$

  where $u_\mu \ge 0$ and $v_\mu \le 0$ denote a super- or sub-solution as in $(H1)$, respectively;

• (H4) there exist $m\in L^\infty (\Omega )$ and $\delta _1>0$ such that $|\{x\in \Omega \,;\,m(x)>0\}|>0$ and

  $$\begin{aligned} f(x,u)u\ge m(x)|u|^p \quad {\text{ for} \text{ every}}\ |u|\le \delta _1, \ {\text{ a.e.}}\ x\in \Omega ; \end{aligned}$$

• (H5) there exist $T^-<0<T^+$ such that $f(x,T^+)\le 0\le f(x,T^-)$ for a.e. $x\in \Omega$;

• (H6) there exist $\delta _2>0$ and an open subset $\Omega _3$ of $\Omega$ such that $f(x,u)u>0$ for every $0<|u|<\delta _2$, a.e. $x\in \Omega _3$.

Theorem 1

Assume $(H1),\, (H2)$ and $(H3)$ or $(H4)$. In addition, in the case of $(H4)$, we also suppose that $u_\mu \in {\mathrm{int}}\,P_B\cup \mathrm{int}\, P$ and $v_\mu \in -{\mathrm{int}}\,P_B\cup -{\mathrm{int}}\,P$ for a super-solution $u_\mu$ and a sub-solution $v_\mu$ as in $(H1)$. Then, for sufficiently large $\mu ,\,(P)_\mu$ has a positive solution $w_{\mu ,1}\in {\mathrm{int}}P_B$, a negative solution $w_{\mu ,2} \in -\mathrm{int\,}P_B$ and a sign-changing solution $w_{\mu ,3}\in X_B\setminus (P_B\cup -P_B)$ with $w_{\mu ,i}\in [v_\mu ,u_\mu ]$ for $i=1,2,3$, where

$$\begin{aligned}&{\mathrm{int}}\, P_B=\left\{ u\in C^1_0(\overline{\Omega })\,;\, u>0 \ \text{ in}\ \Omega \quad {\text{ and}}\quad \frac{\partial u}{\partial \nu }<0\ {\text{ on}}\ \partial \Omega \,\right\} \\ {\text{ and}} \quad&{\mathrm{int}}\, P_B=\{u\in C^1(\overline{\Omega })\,;\, u>0\ \text{ on}\ \overline{\Omega }\,\} \end{aligned}$$

in the Dirichlet case ($Bu=u$) or the Neumann case ($Bu=\partial u/\partial \nu$), respectively.

By applying the above theorem, we have the following result.

Corollary 2

If one of the following conditions holds

• (i) $(H2)$, $(H4)$ and $(H5)$;

• (ii) $(H2)$, $(H5)$ and $(H6)$;

• (iii) $Bu=u$, $(H4)$ and ${\mathrm{ess}}\sup _{x\in \Omega } \,\limsup _{|u|\rightarrow \infty }\frac{f(x,u)}{|u|^{p-2}u}\le 0$,

then for sufficiently large $\mu ,\, (P)_\mu$ has a positive solution $w_{\mu , 1}\in \mathrm{int\,}P_B$, a negative solution $w_{\mu ,2} \in -{\mathrm{int}}P_B$ and a sign-changing solution $w_{\mu ,3}\in X_B\setminus (P_B\cup -P_B)$.

1.2 The existence result in special cases

In this subsection, we state the existence result in the case where the nonlinearity contains a concave term near zero. Precisely, we consider the problem

$$\begin{aligned} \left\{ \begin{array}{ll} -{\mathrm{div}}\, A(x, \nabla u) = \psi _1(\mu )f_1(x,u)+\psi _2(\mu )f_2(x,u)&{\text{ in}}\ \Omega , \\ Bu=0&{\text{ on}}\ \partial \Omega , \end{array}\right. \quad {(\widetilde{{P}})_\mu } \end{aligned}$$

where $\psi _1$ and $\psi _2$ are functions defined on some subset $\fancyscript{M}\subset \mathbb R$ ($\psi _1$ can be identically zero). Here, we suppose that $f_1$ and $f_2$ satisfy $(f)$.

Theorem 3

Assume that

• $(\widetilde{H1})$   for each $\mu \in \fancyscript{M}$, there exist a super-solution $u_\mu \in {\mathrm{int}}\,P_B\cup {\mathrm{int}}\,P$ and a sub-solution $v_\mu \in - {\mathrm{int}}\,P_B\cup - \mathrm{int}\,P$ of ${(\widetilde{{P}})_\mu }$;

• $(\widetilde{H2})$   there exist $D_1>0$ and $\delta _0>0$ such that $f_1(x,u)u\ge -D_1|u|^p$ for every $|u|<\delta _0$ and a.e. $x\in \Omega$;

• $(\widetilde{H3})$   there exists $1<\beta <p$ such that

  $$\begin{aligned} {\mathrm{ess}}\inf _{x\in \Omega } \liminf _{u\rightarrow 0}\frac{f_2(x,u)}{|u|^{\beta -2}u}>0; \end{aligned}$$

• $(\widetilde{H4})$   $\psi _1(\mu )\ge 0$ and $\psi _2(\mu )>0$ for every $\mu \in \fancyscript{M}$.

Then, for every $\mu \in \fancyscript{M},\, (\widetilde{P})_\mu$ has a positive solution $w_{\mu ,1}\in \mathrm{int\,}P_B$, a negative solution $w_{\mu ,2} \in -\mathrm{int\,}P_B$ and a sign-changing solution $w_{\mu ,3}\in X_B\setminus (P_B\cup -P_B)$ with $w_{\mu ,i}\in [v_\mu ,u_\mu ]$ for $i=1,2,3$.

Since any solutions are super- and sub-solutions, the following result follows from Theorem 3.

Corollary 4

Let $\fancyscript{M}=\{\mu \}$ for some $\mu \in \mathbb R$. Assume $(\widetilde{H2}),\, (\widetilde{H3})$ and $(\widetilde{H4})$. If $(\widetilde{P})_\mu$ has two solutions $u\in {\mathrm{int}}\,P_B$ and $v\in -{\mathrm{int}}\,P_B$, then $(\widetilde{P})_\mu$ has at least one sign-changing solution within the order interval $[v,u]$.

Moreover, if we suppose the additional hypothesis that $f_1$ and $f_2$ are odd in the second variable, then the existence of a solution belonging to ${\mathrm{int}}\,P_B$ ensures a pair of sign-changing solutions.

Example 1

In the following cases 1–3, it is known that there exists a positive solution (in ${\mathrm{int}}\,P_B$) for sufficiently small $\mu >0$ of the $p$-Laplace equation

$$\begin{aligned} -\Delta _p u= \psi _1(\mu )f_1(x,u)+\psi _2(\mu )f_2(x,u) \quad {\text{ in}}\ \Omega , \quad Bu=0 \quad {\text{ on}}\ \partial \Omega . \end{aligned}$$

Thus, according to Corollary 4, there exists a sign-changing solution in each case for sufficiently small $\mu >0$ since $f_1$ and $f_2$ are odd in the second variable.

Dirichlet problem:

• 1. $\psi _1(\mu )\equiv 1$, $\psi _2(\mu )=\mu$, $f_1(x,u)=|u|^{q-2}u$, $f_2(x,u)=|u|^{\beta -2}u$ with $1<\beta <p<q\le p^*$ (that is, ABC problem for the $p$-Laplacian);

• 2. $\psi _1(\mu )=\mu$, $\psi _2(\mu )\equiv 1$, $f_1(x,u)=a(x)|u|^{q-2}u$, $f_2(x,u)=|u|^{\beta -2}u-m(x)|u|^{p-2}u$, where $0\not =m\in L^\infty (\Omega )_+$, $a\in C(\overline{\Omega })$ and $2<\beta <p<q<p^*$. Refer to [20] and [1], respectively. Neumann problem:

• 3. $\psi _1(\mu )=\mu ,\, \psi _2(\mu )\equiv 1,\,f_1(x,u)=m(x)|u|^{p-2}u,\, f_2(x,u)=|u|^{\beta -2}u$ with $1<\beta <p$, where $m\in L^\infty (\Omega )$ is a sign-changing function satisfying $\int _{\Omega } m(x)\,\mathrm{d}x\not =0$ (see [8]).

Corollary 5

Assume $(\widetilde{H2})$, $(\widetilde{H3})$ and $(\widetilde{H4})$. Set

$$\begin{aligned} \widetilde{f}(x,u,\mu )=\psi _1(\mu )f_1(x,u)+\psi _2(\mu )f_2(x,u). \end{aligned}$$

Suppose that

• (i) for every $\mu \in \fancyscript{M}$ there exist $T(\mu )_{-}<0<T(\mu )_{+}$ such that $\widetilde{f}(x,T(\mu )_{+},\mu )\le 0\le \widetilde{f}(x,T(\mu )_{-},\mu )$ for a.e. $x\in \Omega$;

or

• (ii) $Bu=u$ and for every $\mu \in \fancyscript{M}$,

  $$\begin{aligned} \mathrm{ess}\sup _{x\in \Omega } \limsup _{|u|\rightarrow \infty }\frac{\widetilde{f}(x,u,\mu )}{|u|^{p-2}u} < \frac{C_0 \lambda _1}{p-1}, \end{aligned}$$

where $\lambda _1>0$ denotes the first eigenvalue of $-\Delta _p$ under the Dirichlet boundary condition. Then, for every $\mu \in \fancyscript{M}$, $(\widetilde{P})_\mu$ has a positive solution $w_{\mu , 1}\in \mathrm{int\,}P_B$, a negative solution $w_{\mu ,2} \in -\mathrm{int\,}P_B$ and a sign-changing solution $w_{\mu ,3}\in X_B\setminus (P_B\cup -P_B)$.

2 The properties of the map $A$

In what follows, the norm on $W_B$ is given by $\Vert u\Vert ^p:=\Vert \nabla u\Vert _p^p+\Vert u\Vert _p^p$, where $\Vert u\Vert _q$ denotes the usual norm of $L^q(\Omega )$ for $u\in L^q(\Omega )$ ($1\le q\le \infty$). Setting

$$\begin{aligned} G(x,y):=\int \limits _0^{|y|} a(x,t)t\,\mathrm{d}t, \end{aligned}$$

(2)

we can easily see that

$$\begin{aligned} \nabla _y G(x,y)=A(x,y) \quad \mathrm{and} \quad G(x,0)=0 \end{aligned}$$

for every $x\in \overline{\Omega }$ (see [27] for details).

Remark 6

the following assertions hold under condition $(A)$:

• (i) for all $x\in \overline{\Omega }$, $A(x,y)$ is maximal monotone and strictly monotone in $y$;

• (ii) $|A(x,y)| \le \frac{C_1}{p-1}|y|^{p-1}$ for every $(x,y)\in \overline{\Omega }\times \mathbb R ^N$;

• (iii) $A(x,y) y \ge \frac{C_0}{p-1} |y|^{p}$ for every $(x,y)\in \overline{\Omega }\times \mathbb R ^N$;

• (iv) $G(x,y)$ is convex in $y$ for all $x$ and satisfies the following inequalities:

  $$\begin{aligned} A(x,y)y \ge G(x,y) \ge \frac{C_0}{p(p-1)}|y|^p \quad \mathrm{and}\quad G(x,y) \le \frac{C_1}{p(p-1)}|y|^p \end{aligned}$$

  (3)

  for every $(x,y)\in \overline{\Omega }\times \mathbb R ^N$,

where $C_0$ and $C_1$ are the positive constants in $(A)$.

Lemma 7

[14, Lemma 2.1.] The map $A$ satisfies the following inequalities:

• (i) $\left|A(x,y)-A(x,y^\prime ) \right| \le c_1 (|y|+|y^\prime |)^{p-2}|y-y^\prime |$;

• (ii) $\left(A(x,y)-A(x,y^\prime )\right)\cdot (y-y^\prime ) \ge c_2 (|y|+|y^\prime |)^{p-2}|y-y^\prime |^2$    if $|y|+|y|^\prime >0$;

• (iii) $\left|A(x,y)-A(x,y^\prime ) \right| \le c_3 |y-y^\prime |^{p-1}$    if $1<p\le 2$;

• (iv) $\left(A(x,y)-A(x,y^\prime )\right)\cdot (y-y^\prime ) \ge c_4|y-y^\prime |^p$    if $p\ge 2$

for every $y$, $y^\prime \in \mathbb R ^N$ and $x\in \Omega$, where $c_i$ is a positive constant ($i=1,2,3,4$).

The following result is important for the proof of the Palais–Smale condition for the functionals related to our problem.

Proposition 8

[27, Proposition 1] Let $V:W_B \rightarrow W_B^*$ be the map defined by

$$\begin{aligned} \langle V(u), v \rangle =\int \limits _{\Omega }A(x,\nabla u) \nabla v\,\mathrm{d}x \end{aligned}$$

for $u,\, v\in W_B$. Then, $V$ has the $(S)_+$ property, that is, any sequence $\{u_m\}$ weakly convergent to $u$ strongly converges to $u$ provided $\limsup _{m\rightarrow \infty } \langle V(u_m), u_m -u \rangle \le 0$.

Proposition 9

For $\lambda >0$, we define a map $T_\lambda :W_B\rightarrow W_B^*$ by

$$\begin{aligned} \langle T_\lambda (u),v\rangle =\int \limits _\Omega A(x,\nabla u)\nabla v\ \mathrm{d}x + \lambda \int \limits _\Omega |u|^{p-2}uv\ \mathrm{d}x \end{aligned}$$

(4)

for $u,\, v\in W_B$. Then, the inverse $T_\lambda ^{-1} :W_B^* \rightarrow W_B$ of $T_\lambda$ exists and it is continuous.

Proof

We have that $T_\lambda$ is injective due to the monotonicity of $A$ and $|u|^{p-2}u$. Furthermore, $T_\lambda$ is continuous being the potential operator of a $C^1$-function and is coercive on the basis of Remark 6 (iii). Since $T_\lambda$ is monotone, hemicontinuous and coercive, we infer that $T_\lambda$ is surjective (see [32, p. 557]). So, there exists the inverse operator $T_\lambda ^{-1}:W_B^*\rightarrow W_B$, which is known to be strictly monotone, semicontinuous and bounded (see [32, p. 557]).

We show that $T_\lambda ^{-1}:W_B^*\rightarrow W_B$ is continuous. Let $\xi _n\rightarrow \xi$ in $W_B^*$. There exists a unique $u_n\in W_B$ such that $T_\lambda (u_n)=\xi _n$. This ensures that

$$\begin{aligned} \min \left\{ \frac{C_0}{p-1},\lambda \right\} \Vert u_n\Vert ^p&\le \int \limits _\Omega A(x,\nabla u_n)\nabla u_n\ \mathrm{d}x + \lambda \int \limits _\Omega |u_n|^p\ \mathrm{d}x\\&= \langle \xi _n,u_n\rangle \le \Vert \xi _n\Vert _{W_B^*}\Vert u_n\Vert , \end{aligned}$$

thereby the sequence $\{u_n\}$ is bounded in $W_B$. Hence, along a relabeled subsequence, we may suppose that $u_n\rightharpoonup u$ in $W_B$ and $u_n\rightarrow u$ in $L^p(\Omega )$ as $n\rightarrow \infty$, with some $u\in W_B$. Passing to the limit in the equality

$$\begin{aligned} \int \limits _\Omega A(x,\nabla u_n)\nabla (u_n-u)\ \mathrm{d}x + \lambda \int \limits _\Omega |u_n|^{p-2}u_n(u_n-u)\ \mathrm{d}x = \langle \xi _n,u_n-u\rangle , \end{aligned}$$

we find that

$$\begin{aligned} \lim _{n\rightarrow \infty }\int \limits _\Omega A(x,\nabla u_n)\nabla (u_n-u)\ \mathrm{d}x=0. \end{aligned}$$

Recalling that the operator $V:W_B\rightarrow W_B^*$ satisfies the $(S)_+$-property (refer to Proposition 8), it turns out that $u_n\rightarrow u$ strongly in $W_B$ (for the whole sequence $\{u_n\}$). Hence, $\xi _n=T_\lambda (u_n)\rightarrow T_\lambda (u)$, which leads to $\xi =T_\lambda (u)$. Therefore, we have that $T_\lambda ^{-1}(\xi _n)\rightarrow T_\lambda ^{-1}(\xi )$. We conclude that $T_\lambda ^{-1}$ is continuous.$\square$

The following result is proved by an argument similar to [22, Lemma 4.6.]. We give the proof in “Appendix”.

Proposition 10

Assume $\lambda >0$, $1<p\le N$ and $r>N/p$. Let $T_\lambda$ be the map defined by (4). Then, there exists a $D_2>0$ such that

$$\begin{aligned} \Vert T_\lambda ^{-1}(u)\Vert _\infty \le D_2\Vert u\Vert _r^{1/(p-1)} \quad \mathrm{for\ every}\ u\in L^r(\Omega ), \end{aligned}$$

so, $T_\lambda ^{-1}$ maps a bounded set of $L^r(\Omega )$ into a bounded set of $L^\infty (\Omega )$.

3 The existence of critical points via descending flow

Throughout this section, we suppose that $h:\Omega \times \mathbb R \rightarrow \mathbb R$ is a Carathéodory function satisfying $h(x,0)=0$ a.e. $x\in \Omega$ and there exist $C>0$ such that

$$\begin{aligned} |h(x,t)| \le C(1+|t|^{p-1}) \quad \mathrm{for\ every}\ t\in \mathbb R , \ \mathrm{a.e.}\ x\in \Omega , \end{aligned}$$

(5)

Under (5), we define a $C^1$ functional $J$ on $W_B$ by

$$\begin{aligned} J(u)&:= \int \limits _{\Omega }G(x,\nabla u)\,\mathrm{d}x -\int \limits _{\Omega }\int \limits _0^{u(x)} h(x,t)\,\mathrm{d}t\mathrm{d}x.\\ \mathrm{There\ holds}\quad \langle J^\prime (u),v\rangle&= \int \limits _{\Omega }A(x,\nabla u)\nabla v\,\mathrm{d}x-\int \limits _{\Omega }h(x,u)v\,\mathrm{d}x \end{aligned}$$

for $u$, $v\in W_B$.

Theorem 11

Assume the following conditions:

• (A1) there exists $\lambda _0>0$ such that

  $$\begin{aligned} h(x,u)u+\lambda _0 |u|^p\ge 0 \quad \mathrm{for\ every}\ u\in \mathbb R , \ \mathrm{a.e.}\ x\in \Omega ; \end{aligned}$$

• (A2) there exists $\gamma \in C([0,1]\,,\,X_B)$ such that $\gamma (0)\in P_B$, $\gamma (1)\in -P_B$ and $\max _{t\in [0,1]} J(\gamma (t))<0$.

If $J$ is coercive on $W_B$, then $J$ has at least three critical points $w_1\in \mathrm{int\,}P_B$, $w_2 \in -\mathrm{int\,}P_B$ and $w_3\in X_B\setminus (P_B\cup -P_B)$.

Set

$$\begin{aligned} \varphi _p(u):=|u|^{p-2}u \quad \mathrm{and} \quad B_\lambda (u):=T_\lambda ^{-1}(h(\cdot ,u)+\lambda \varphi _p(u)) \end{aligned}$$

(6)

for $u\in W_B$ and $\lambda >0$, where $T_\lambda ^{-1}$ is the inverse of $T_\lambda$ (see Proposition 9 for the existence of the inverse). Note that under $N>p$, $h(\cdot ,u)+\varphi _p(u)\in L^{p^*/(p^*-1)}(\Omega )$ provided $u\in L^{p^*(p-1)/(p^*-1)}(\Omega )$ and so $B_\lambda :L^{p^*(p-1)/(p^*-1)}(\Omega )\rightarrow W_B$ is well defined. Also, we note that $B_\lambda :W_B\rightarrow W_B$ is continuous according to Proposition 9 and (5).

Throughout this section, we denote the critical set of $J$ by $K$, that is, $K:=\{u\in W_B\,;\,J^\prime (u)=0\,\}$.

Remark 12

If $u\in W_B\cap L^\infty (\Omega )$, then $v=B_\lambda (u)\in C^{1,\alpha }(\overline{\Omega })$ (some $\alpha \in (0,1)$) and $v$ is a solution of

$$\begin{aligned} -\mathrm{div}\,A(x,\nabla v)+\lambda |v|^{p-2}v=h(x,u)+\lambda |u|^{p-2}u \quad \mathrm{in}\ \Omega ,\quad Bv=0\ \mathrm{on}\ \partial \Omega . \end{aligned}$$

Indeed, $v=B_\lambda (u)$ satisfies

$$\begin{aligned} \int \limits _{\Omega }A(x,\nabla v)\nabla w\,\mathrm{d}x+\lambda \int \limits _{\Omega }|v|^{p-2}vw\,\mathrm{d}x=\int \limits _{\Omega }h(x,u)w\,\mathrm{d}x +\lambda \int \limits _{\Omega }|u|^{p-2}uw\,\mathrm{d}x \end{aligned}$$

for every $w\in W_B$. Because of $u\in L^\infty (\Omega )$, we have $v\in L^\infty (\Omega )$ by the Moser iteration process or Lemma 13 in the next subsection. Therefore, we see that $v\in C^{1,\alpha }(\overline{\Omega })$ ($0<\alpha <1$) by the regularity result in [23].

In the case of the Neumann problem, by [13, Theorem 3], $v$ satisfies the boundary condition

$$\begin{aligned} 0=\frac{\partial v}{\partial \nu _A}=A(\cdot , \nabla v) \, \nu =a(\cdot , |\nabla v|)\frac{\partial v}{\partial \nu } \quad \mathrm{in}\ W^{-1/q,q}(\partial \Omega ) \end{aligned}$$

for every $1<q<\infty$ (see [13] for the definition of $W^{-1/q,q}(\partial \Omega )$). Since $v\in C^{1,\alpha }(\overline{\Omega })$ and $a(x,t)>0$ for every $t\not =0$, $v$ satisfies the Neumann boundary condition, that is, $\frac{\partial v}{\partial \nu }(x)=0$ for every $x\in \partial \Omega$.

Consequently, the critical points of $J$ correspond to the fixed points of $B_\lambda$.

Moreover, we remark that $K\subset X_B$. In fact, if $u$ is a critical point of $J$, then we have $u\in L^\infty (\Omega )$ by the Moser iteration process (refer to Theorem C in [26]). Thus, from the above argument, $u\in X_B$ follows.

3.1 Constructing a descending flow

Lemma 13

For $\lambda >0$, $B_\lambda$ satisfies the following properties:

• (i) there exists a $D_3=D_3(\lambda )>0$ such that

  $$\begin{aligned} \Vert B_\lambda (u)\Vert _\infty \le D_3(\Vert u\Vert _\infty +1) \quad \mathrm{for\ every}\ u\in L^\infty (\Omega ); \end{aligned}$$

• (ii) for every $R>0$, there exist $\alpha =\alpha (R,\lambda )\in (0,1)$ and $M=M(R,\lambda )>0$ such that $\Vert B_\lambda (u)\Vert _{C^{1,\alpha }_B(\overline{\Omega })}\le M$ for all $u\in L^\infty (\Omega )$ with $\Vert u\Vert _\infty \le R$;

• (iii) If $N\ge p$ and $r>\max \{N/p,1/(p-1)\}$, then there exists a $D_4=D_4(\lambda )>0$ such that

  $$\begin{aligned} \Vert B_\lambda (u)\Vert _\infty \le D_4(\Vert u\Vert _{r(p-1)}+1) \quad \mathrm{for\ every}\ u\in L^{r(p-1)}(\Omega ). \end{aligned}$$

Proof

• (i) Let $u\in L^\infty (\Omega )$ and $v=B_\lambda (u)(\in W_B)$. First, we consider $N<p$. Because $W_B\hookrightarrow L^\infty (\Omega )$ is continuous, there exists $D>0$ such that $D\Vert w\Vert _\infty \le \Vert w\Vert$ for every $w\in W_B$. Hence, by Hölder’s inequality and (5), we obtain

  $$\begin{aligned}&\min \left\{ \frac{C_0}{p-1},\lambda \right\} D^{p-1}\Vert v\Vert _\infty ^{p-1}\Vert v\Vert \le \min \left\{ \frac{C_0}{p-1},\lambda \right\} \Vert v\Vert ^p \\&\le \int \limits _{\Omega }A(x,\nabla v)\nabla v\,\mathrm{d}x+\lambda \int \limits _{\Omega }|v|^p\,\mathrm{d}x =\langle T_\lambda (v), v\rangle =\langle h(\cdot ,u)+\lambda \varphi _p(u),v \rangle \\&\le \Vert v\Vert _p|\Omega |^{p/(p-1)}\left(C+(C+\lambda )\Vert u\Vert _\infty ^{p-1}\right) \le \Vert v\Vert |\Omega |^{p/(p-1)}\left(C+(C+\lambda )\Vert u\Vert _\infty ^{p-1}\right). \end{aligned}$$

  This proves our conclusion. In the case of $N\ge p$, we choose $r>N/p$. Then, by Proposition 10 and (5), we have

  $$\begin{aligned} \Vert v\Vert _\infty \!\le \! D_2\Vert h(\cdot ,u)\!+\!\lambda \varphi _p(u)\Vert _r^{1/(p\!-\!1)} \!\le \! 2^{\frac{1}{p\!-\!1}} D_2 \left(C^{\frac{1}{p-1}}\!+\!(\lambda \!+\!C)^{\frac{1}{p-1}}\Vert u\Vert _\infty \right) |\Omega |^{1/(r(p-1))}, \end{aligned}$$

  which establishes (i).

• (ii) This assertion follows from (i) and the argument in Remark 12. Note that $\alpha$ and $M$ depend on $\Vert u\Vert _\infty$ generally (refer to [23]).

• (iii) By Proposition 10 and (5), we obtain

  $$\begin{aligned} \Vert B_\lambda (u)\Vert _\infty&\le D_2\Vert h(\cdot ,u)+\lambda \varphi _p(u)\Vert _r^{1/(p-1)} \le D_2 C^\prime (1+(\lambda +1)\Vert u\Vert _{r(p-1)}) \end{aligned}$$

  for every $u\in L^{r(p-1)}(\Omega )$, where $C^\prime >0$ is a constant independent of $u$.

$\square$

Lemma 14

Let $p<N$. Define inductively

$$\begin{aligned} q_0:=p^* \quad \mathrm{and}\quad q_{n+1}:=p^*q_n/p=(p^*/p)^{n+1}p^*. \end{aligned}$$

(7)

For every $n\in \mathbb N \cup \{0\}$, there exists a positive constant $C^*_{n+1}$ such that

$$\begin{aligned} \Vert B_\lambda (u)\Vert _{q_{n+1}} \le C_{n+1}^* (1+\Vert u\Vert _{q_n}) \quad \mathrm{for\ every}\ u\in L^{q_n}(\Omega ). \end{aligned}$$

Proof

Let $u\in L^{q_n}(\Omega )$ and $v=B_\lambda (u)\in W_B$. Now, for $\varphi =v_+$ or $v_-$ and $M>0$, we put $\varphi _M(x):=\min \,\{\varphi (x), M\}$. By taking $\varphi _M^{q+1}$ (if $\varphi =u_+$) or $-\varphi _M^{q+1}$ (if $\varphi =u_-$) with $q>0$ as test function in $T_\lambda (v)=h(\cdot ,u)+\lambda |u|^{p-2}u$ in $W_B^*$, (note that $\nabla (\varphi _M^{q+1})=(q+1)\varphi _M^q \nabla \varphi _M$), we have

$$\begin{aligned}&\frac{C_0(q+1)}{p-1} \int \limits _{\Omega }|\nabla \varphi _M|^p \varphi _M^q \,\mathrm{d}x +\lambda \Vert \varphi _M\Vert _{p+q}^{p+q} \nonumber \\&\le (q+1)\,\int \limits _{\Omega }\varphi _M^{q}\, A(x,\pm \nabla \varphi _M)(\pm \nabla \varphi _M)\,\mathrm{d}x +\lambda \Vert \varphi _M\Vert _{p+q}^{p+q} \nonumber \\&\le \int \limits _{\Omega }(C+(C+\lambda )|u|^{p-1})|\varphi _M|^{q+1}\,\mathrm{d}x \nonumber \\&\le C^\prime (1+(1+\lambda )\Vert u\Vert _{q_n}^{p-1}) \Vert \varphi _M\Vert _{(q+1)q_n/(q_n-p+1)}^{q+1}, \end{aligned}$$

(8)

where we use (5), Hölder’s inequality and Remark 6 (iii). On the other hand, the embedding of $W_B$ into $L^{p^*}(\Omega )$ guarantees the existence of $C_*>0$ satisfying

$$\begin{aligned} C_*\Vert \varphi _M\Vert _{p^*(p+q)/p}^{p+q}&= C_*\Vert (\varphi _M)^{1+q/p}\Vert _{p^*}^p \le \Vert (\varphi _M)^{1+q/p}\Vert ^p \nonumber \\&= (1+q/p)^p \int \limits _{\Omega }|\nabla \varphi _M|^p\,(\varphi _M)^q\,\mathrm{d}x+\Vert \varphi _M\Vert _{p+q}^{p+q}. \end{aligned}$$

(9)

Combining (8) and (9), it follows that

$$\begin{aligned} C_*\Vert \varphi _M\Vert _{p^*(p+q)/p}^{p+q} \le D (1+\Vert u\Vert _{q_n}^{p-1})\Vert \varphi _M\Vert _{(q+1)q_n/(q_n-p+1)}^{q+1}, \end{aligned}$$

where $D=D(\lambda ,p,q,|\Omega |)$ is a positive constant independent of $u$ and $v$. Here, we choose $q=q_n-p$. Then, we obtain

$$\begin{aligned} C_*\Vert \varphi _M\Vert _{q_{n+1}}^{p-1}=C_*\Vert \varphi _M\Vert _{p^*q_n/p}^{p-1} \le D(1+\Vert u\Vert _{q_n}^{p-1})|\Omega |^{(p^*-p)(q_n-p+1)/p^*q_n} \end{aligned}$$

because of $q_n<q_np^*/p=q_{n+1}$ and using Hölder’s inequality. Therefore, by letting $M\rightarrow +\infty$, we see that $v_\pm \in L^{q_{n+1}}(\Omega )$ and $\Vert v_\pm \Vert _{q_{n+1}}\le D^\prime (1+\Vert u\Vert _{q_n})$ for some positive constant $D^\prime =D^\prime (\lambda ,p,q_n,|\Omega |,C_*)$.$\square$

Lemma 15

Assume $(A1)$ and let $\lambda _0>0$ be a constant as in $(A1)$. Then, for each $\lambda > \lambda _0$, we have $B_{\lambda }(u)\in \pm \,\mathrm{int}\,P_B$ provided $u\in \pm \, \widetilde{P}_B\cap L^\infty (\Omega )\setminus \{0\}$, respectively.

Proof

We may assume that $u\in \widetilde{P}_B\cap L^\infty (\Omega )\setminus \{0\}$ by considering $-u$ in the other case. Let $u\in \widetilde{P}_B\cap L^\infty (\Omega )\setminus \{0\}$ and $v=B_\lambda (u)$. Then, $v\in C^{1,\alpha }(\overline{\Omega })$ (some $\alpha \in (0,1)$) and $v$ is a solution of

$$\begin{aligned} -\mathrm{div}\,A(x,\nabla v)+\lambda |v|^{p-2}v=h(x,u)+\lambda u^{p-1} \quad \mathrm{in}\ \Omega ,\quad Bv=0\ \mathrm{on}\ \partial \Omega \end{aligned}$$

(see Remark 12). By taking $-v_-$ as a test function, we obtain

$$\begin{aligned} \min \left\{ \frac{C_0}{p-1},\lambda \right\} \Vert v_-\Vert ^p&\le \int \limits _{\Omega }A(x,\nabla v)(-\nabla v_-)\,\mathrm{d}x +\lambda \int \limits _{\Omega }v_-^p\,\mathrm{d}x\\&= -\int \limits _{\Omega }(h(x,u)+\lambda u^{p-1})\,v_-\,\mathrm{d}x\le 0 \end{aligned}$$

since $h(x,u)+\lambda u^{p-1}\ge 0$ holds by $(A1)$, whence $v_-=0$ a.e. $\Omega$. Furthermore, we note that $h(\cdot ,u)+\lambda u^{p-1}\not =0$ in $W_B^*$ by the inequality

$$\begin{aligned} \langle h(\cdot ,u)+\lambda u^{p-1}, u\rangle&= \int \limits _{\Omega }(h(x,u)+\lambda _0 u^{p-1})u\,\mathrm{d}x+(\lambda -\lambda _0)\int \limits _{\Omega }u^p\,\mathrm{d}x\\&\ge (\lambda -\lambda _0)\int \limits _{\Omega }u^p\,\mathrm{d}x>0 \end{aligned}$$

(note $u\not =0$). This yields $v\not =0$ because of $v=B_\lambda (u)=T_\lambda ^{-1}(h(\cdot ,u)+\lambda \varphi _p(u))$.

By noting that $v\in C^{1,\alpha }(\overline{\Omega })$ and $v\not =0$, we have $v(x)>0$ for every $x\in \Omega$ by Theorem B in [26]. In addition, due to the strong maximum principle (see Theorem A in [26]), we easily see that $\partial v(x)/\partial \nu <0$ for every $x\in \partial \Omega$ provided $v(x)=0$. Hence, under the Neumann boundary condition (that is, $Bv=\partial v/\partial \nu =0$), $v(x)>0$ for every $x\in \overline{\Omega }$. This means that $v\in \mathrm{int}\, P_B$.$\square$

The proof of the following lemma can be shown by the argument in [5, Lemma 3.7 and 3.8] and Lemma 7. Thus, we omit the proof.

Lemma 16

Let $\lambda >0$. Then, there exist $d_i=d_i(\lambda )>0$ $(1\le i\le 4)$ such that

• (i) $\langle J^\prime (u), u-B_\lambda (u) \rangle \ge d_1\Vert u-B_\lambda (u)\Vert ^2(\Vert u\Vert +\Vert B_\lambda (u)\Vert )^{p-2}$    if $1<p\le 2$;

• (ii) $\langle J^\prime (u), u-B_\lambda (u) \rangle \ge d_2\Vert u-B_\lambda (u)\Vert ^p$    if $p\ge 2$;

• (iii) $\Vert J^\prime (u)\Vert _{W_B^*}\le d_3\Vert u-B_\lambda (u)\Vert ^{p-1}$    if $1<p\le 2$;

• (iv) $\Vert J^\prime (u)\Vert _{W_B^*}\le d_4\Vert u-B_\lambda (u)\Vert (\Vert u\Vert +\Vert B_\lambda (u)\Vert )^{p-2}$    if $p\ge 2$

for every $u\in W_B$, where $B_\lambda$ is the operator defined by (6).

The next result follows from a similar argument as in [5, Lemma 4.1.] using the properties of $B_\lambda$ described in Lemmas 16, 15, 14 and 13.

Lemma 17

Let $\lambda > \lambda _0 \,(\lambda _0$ being the positive constant as in $(A1)$). Then, there exists a locally Lipschitz continuous operator $V_\lambda$ from $X_B\setminus K$ into $X_B$ such that

• (i) For $u\in X_B\setminus K$, we have

  $$\begin{aligned}&\langle J^\prime (u), u-V_\lambda (u) \rangle \ge \frac{d_1}{2}\Vert u-B_\lambda (u)\Vert ^2(\Vert u\Vert +\Vert B_\lambda (u)\Vert )^{p-2} \quad \mathrm{if}\ 1<p\le 2;\\&\langle J^\prime (u), u-V_\lambda (u) \rangle \ge \frac{d_2}{2}\Vert u-B_\lambda (u)\Vert ^p \quad \mathrm{if}\ p\ge 2;\\&\frac{1}{2}\Vert u-B_\lambda (u)\Vert \le \Vert u-V_\lambda (u)\Vert \le 2\Vert u-B_\lambda (u)\Vert ; \end{aligned}$$

  where $d_1$ and $d_2$ are the positive constants in Lemma 16.

• (ii) $V_\lambda (u)\in \pm \, \mathrm{int}\,P_B$ for every $u\in \pm \,P_B \setminus K$, respectively;

• (iii) For the sequence $\{q_n\}$ defined by (7), $V_\lambda$ satisfies

  $$\begin{aligned} \Vert V_\lambda (u)\Vert _{q_{n+1}} \le C_{n+1}^* (2+|\Omega |+\Vert u\Vert _{q_n}) \quad \text{ for} \text{ every}\ u\in X_B\setminus K, \end{aligned}$$

  where $C_{n+1}^*$ is the positive constant obtained in Lemma 14.

• (iv) If $N\ge p$ and $r>\max \{N/p,1/(p-1)\}$, then $V_\lambda$ satisfies

  $$\begin{aligned} \Vert V_\lambda (u)\Vert _\infty \le D_4(\Vert u\Vert _{r(p-1)}+2+|\Omega |) \quad \text{ for} \text{ every}\ u\in X_B\setminus K, \end{aligned}$$

  where $D_4$ is the positive constant obtained in Lemma 13 (iii) for $r$ above;

• (v) there holds

  $$\begin{aligned} \Vert V_\lambda (u)\Vert _\infty \le D_3(2+\Vert u\Vert _\infty ) \quad \text{ for} \text{ every}\ u\in X_B\setminus K, \end{aligned}$$

  where $D_3$ is the positive constant obtained in Lemma 13 (i);

• (vi) for every $R>0$, there exist $\alpha \in (0,1)$ and $M>0$ such that $\Vert V_\lambda (u)\Vert _{C^{1,\alpha }_B(\overline{\Omega })}\le M$ for every $u\in X_B\setminus K$ with $\Vert u\Vert _\infty \le R$.

Proof

For $u\in W_B\setminus K$, we define

$$\begin{aligned} \delta _1(u)&:= \frac{1}{2}\Vert u-B_\lambda (u)\Vert ,\\ \delta _2(u)&:= \frac{d_1}{2d_3}\Vert u-B_\lambda (u)\Vert ^{3-p} (\Vert u\Vert +\Vert B_\lambda (u)\Vert )^{p-2} \quad \mathrm{if}\ 1<p\le 2,\\ \delta _2(u)&:= \frac{d_2}{2d_4}\Vert u-B_\lambda (u)\Vert ^{p-1} (\Vert u\Vert +\Vert B_\lambda (u)\Vert )^{2-p} \quad \mathrm{if}\ p>2, \end{aligned}$$

where $d_i>0$ ($i = 1,2,3,4$) denotes a constant as in Lemma 16. Note that $\delta _1(u)>0$ and $\delta _2(u)>0$ by $u\not \in K$. We denote the usual $X_B$ norm by $\Vert \cdot \Vert _X$. Choose an open neighborhood $N(u)$ of $u$ in $X_B$ such that

$$\begin{aligned} N(u):=\left\{ \begin{array}{ll} v\in X_B\setminus K\,&;\,\Vert u-v\Vert _X<1/2,\ \delta _1(v)>\delta _1(u)/2,\ \delta _2(v)>\delta _1(u)/2,\, \\&\quad \Vert B_\lambda (u)-B_\lambda (v)\Vert < \min \{\delta _1(u),\delta _2(u)\}/4 \end{array} \right\} . \end{aligned}$$

Note that

$$\begin{aligned} \Vert v-w\Vert _X<1 \quad \mathrm{and} \quad \Vert B_\lambda (v)-B_\lambda (w)\Vert <\min \{\delta _1(v), \delta _1(w), \delta _2(v), \delta _2(w)\} \end{aligned}$$

(10)

for every $v,\, w\in N(u)$. It is obvious that $\{N(u)\,;\,u\in X_B\setminus K\}$ is an open covering of $X_B\setminus K$. According to the paracompactness of $X_B\setminus K$, $X_B\setminus K$ has a locally finite open refinement $\{U_\xi \}_{\xi \in I}$ of $\{N(u)\,;\,u\in X_B\setminus K\}$. If there exists a $U_\xi$ such that $U_\xi \cap P_B\not =\emptyset$ and $U_\xi \cap -P_B\not =\emptyset$, then we replace $U_\xi$ with two open sets $U_\xi \setminus P_B$ and $U_\xi \setminus -P_B$. In this way, we may assume that $U_\xi$ satisfies one of the following conditions:

(a) $U_\xi \cap P_B=\emptyset$ and $U_\xi \cap -P_B=\emptyset$;    (b) $U_\xi \cap P_B\not =\emptyset$ and $U_\xi \cap -P_B=\emptyset$; (c) $U_\xi \cap P_B=\emptyset$ and $U_\xi \cap -P_B\not =\emptyset$.

For each case above, we choose a point $u_\xi$ satisfying $u_\xi \in U_\xi$, $u_\xi \in U_\xi \cap P_B$ and $u_\xi \in U_\xi \cap -P_B$ in the case of (a), (b) and (c), respectively. Let $\{\psi _\xi \}_{\xi \in I}$ be a $C^1$ partition of the unity with respect to $\{U_\xi \}_{\xi \in I}$.

Define

$$\begin{aligned} V_\lambda (u):=\sum _{\xi \in I} \psi _\xi (u) B_\lambda (u_\xi ) \end{aligned}$$

for every $u\in X_B\setminus K$. It is easily shown that $V_\lambda$ is locally Lipschitz continuous since $\{U_\xi \}_{\xi \in I}$ is locally finite.

• (i) For $\xi \in I$ such that $\psi _\xi (u)\not =0$, we have $\Vert B_\lambda (u_\xi )-B_\lambda (u)\Vert <\min \{\delta _1(u),\delta _2(u)\}$ by (10). Therefore, we get $\Vert B_\lambda (u)-V_\lambda (u)\Vert <\delta _1(u)=\Vert u-B_\lambda (u)\Vert /2$, whence $\Vert u-B_\lambda (u)\Vert /2\le \Vert u-V_\lambda (u)\Vert \le 3\Vert u-B_\lambda (u)\Vert /2$ holds. The other inequalities follow from $\Vert B_\lambda (u_\xi )-B_\lambda (u)\Vert <\delta _2(u)$, the inequality for $\Vert J^\prime (\cdot )\Vert _{W_B^*}$ in Lemma 16, (10) and $\langle J^\prime (u), u-V_\lambda (u)\rangle \ge \langle J^\prime (u), u-B_\lambda (u)\rangle -\Vert J^\prime (u)\Vert _{W_B^*}\Vert B_\lambda (u)-V_\lambda (u)\Vert$.

• (ii) Let $u\in P_B\setminus K$. Then, for $\xi \in I$ such that $\psi _\xi (u)\not =0$, we see that $u_\xi \in P_B$ because $U_\xi$ satisfies (b). Thus, $B_\lambda (u_\xi )\in \mathrm{int}\, P_B$ holds by Lemma 15. Since $\mathrm{int}\, P_B$ is convex, our assertion is proved. Similarly, we can show that $V_\lambda (u)\in -\mathrm{int}\, P_B$ provided $u\in (-P_B)\setminus K$.

• (iii) According to Lemma 14, we obtain

  $$\begin{aligned} \Vert V_\lambda (u)\Vert _{q_{n+1}}&\le C_{n+1}^* \sum _{\xi \in I} \psi _\xi (u) (1+\Vert u_\xi \Vert _{q_n}) \\&\le C_{n+1}^*\sum _{\xi \in I} \psi _\xi (u) (1+\Vert u\Vert _{q_n}+\Vert u_\xi -u\Vert _{q_n}) \le C_{n+1}^*(2+|\Omega |+\Vert u\Vert _{q_n}) \end{aligned}$$

  by noting (10), $\Vert u_\xi -u\Vert _{q_n}\le \Vert u_\xi -u\Vert _\infty |\Omega |^{1/q_n}$ and $|\Omega |^{1/q_n}\le 1+|\Omega |$.

• (iv) By a similar argument to (iii) and Lemma 13 (iii), our conclusion holds.

• (v) By a similar argument to (iii) and Lemma 13 (i), our conclusion holds.

• (vi) For $\xi \in I$ such that $\psi _\xi (u)\not =0$, $\Vert u_\xi \Vert _\infty \le \Vert u_\xi -u\Vert _\infty +\Vert u\Vert _\infty \le 1+\Vert u\Vert _\infty$ holds. Thus, if $\Vert u\Vert _\infty \le R$, then by Lemma 13 (ii), there exist $\alpha =\alpha (R+1,\lambda )\in (0,1)$ and $M=M(R+1,\lambda )>0$ with $\Vert B_\lambda (u_\xi )\Vert _{C^{1,\alpha }_B(\overline{\Omega })}\le M$ for $\xi \in I$ such that $\psi _\xi (u)\not =0$. Hence, our conclusion follows.

$\square$

Fix $\lambda > \lambda _0$, where $\lambda _0>0$ is the constant as in $(A1)$. For $u \in X_B\setminus K$, we consider the following initial value problem in $X_B$:

$$\begin{aligned} \left\{ \begin{array}{l} \dfrac{\mathrm{d}\eta }{\mathrm{d}t}(t)=-\eta (t)+V_\lambda \left(\,\eta (t)\right) \\ \eta (0)=u, \end{array} \right. \end{aligned}$$

where $V_\lambda$ is the locally Lipschitz continuous map from $X_B\setminus K$ into $X_B$ constructed in Lemma 17. Let $\eta (t,u)$ be the unique solution of the above problem considered in $X_B$. Moreover, we denote by $[0,\tau (u))$ the right maximal interval of existence of $\eta (t,u)$.

Lemma 18

The following assertions hold:

• (i) $J(\eta (t,u))\le J(u)$ for every $u\in X_B\setminus K$ and $t\in [0,\tau (u))$;

• (ii) If $\tau (u)<+\infty$ and $\eta (t,u)$ weakly converges to some $w$ in $W_B$ as $t\rightarrow \tau (u)-0$, then $w\in K$ and $\eta (t,u)$ converges to $w$ in $X_B$ as $t\rightarrow \tau (u)-0$.

• (iii) if $\eta (t,u)$ converges to some $w$ in $W_B$ as $t\rightarrow \tau (u)-0$, then $w\in X_B$ and $\eta (t,u)$ converges to $w$ in $X_B$ as $t\rightarrow \tau (u)-0$.

Proof

• (i) Let $u\in X_B\setminus K$ and $t\in [0,\tau (u))$. Then, by the property of $V_\lambda$ in Lemma 17 (i), $\mathrm{d}\,J(\eta (t,u))/\mathrm{d}t\le 0$. Thus, our conclusion holds.

• (ii) Let $\tau (u)<+\infty$. Note that $\eta (t,u)$ satisfies

  $$\begin{aligned} \eta (t,u)=e^{-t}u +\int \limits _0^t e^{-t+s} V_\lambda (\eta (s,u))\,\mathrm{d}s \quad \mathrm{for}\ 0\le t<\tau (u) \end{aligned}$$

  in $X_B$. Then, we obtain

  $$\begin{aligned} \Vert \eta (t,u)\Vert _\infty \le e^{-t}\Vert u\Vert _\infty +2D_3 +D_3\int \limits _0^t e^{-t+s}\Vert \eta (s,u)\Vert _\infty \,\mathrm{d}s \quad \mathrm{for}\ 0\le t<\tau (u) \end{aligned}$$

  by Lemma 17 (v). Hence, due to Gronwall’s inequality, we have

  $$\begin{aligned} \Vert \eta (t,u)\Vert _\infty \le e^{D_3\tau (u)}(\Vert u\Vert _\infty +2D_3) \quad \mathrm{for\ every}\ t\in [0,\tau (u)). \end{aligned}$$

  Therefore, it follows from Lemma 17 (vi) that there exists $\alpha \in (0,1)$ such that $\{\int _0^t e^{-t+s} V_\lambda (\eta (s,u))\,\mathrm{d}s\,;\,t\in [0,\tau (u))\}$ is bounded in $C^{1,\alpha }_B(\overline{\Omega })$. Since the embedding of $C^{1,\alpha }_B(\overline{\Omega })$ into $X_B$ is compact and $\eta (t,u)\rightharpoonup w$ in $W_B$ as $t\rightarrow \tau (u)-0$, $\eta (t,u)$ converges to $w$ in $X_B$ as $t\rightarrow \tau (u)-0$. By the definition of $\tau (u)$ and $\tau (u)<+\infty$, we see that $w\in K$ holds.

• (iii) For the proof, it suffices to show that $\{\int _0^t e^{-t+s} V_\lambda (\eta (s,u))\,\mathrm{d}s\,;\,t\in [0,\tau (u))\}$ is bounded in $L^\infty (\Omega )$ proceeding as in (ii). In the case of $p>N$, it is obvious because $W_B\hookrightarrow L^\infty (\Omega )$ is continuous and $\eta (t,u)$ converges to some $w$ in $W_B$.

In the case of $N=p$, then by taking an $r>\max \{1,1/(p-1)\}$, we obtain

$$\begin{aligned} \int \limits _0^t e^{-t+s} \Vert V_\lambda (\eta (s,u))\Vert _\infty \,\mathrm{d}s \le D_4 (2+|\Omega |+\sup _{t\in [0,\tau (u))}\Vert \eta (t,u)\Vert _{r(p-1)}) \end{aligned}$$

due to Lemma 17 (iv) and the continuity of $W_B\hookrightarrow L^{r(p-1)}(\Omega )$ (note that $\{\eta (t,u)\,;\,t\in [0,\tau (u))\}$ is bounded in $W_B$).

Now, we consider the case of $N>p$. Note that we can choose $n_0\in \mathbb N$ such that $q_{n_0}>N(p-1)/p$ since $q_n\rightarrow \infty$ as $n\rightarrow \infty$, where $\{q_n\}$ is the increasing sequence defined by (7). By considering

$$\begin{aligned} W_B \stackrel{V_\lambda }{\longrightarrow } L^{q_0}(\Omega ) \stackrel{V_\lambda }{\longrightarrow } L^{q_1}(\Omega ) \stackrel{V_\lambda }{\longrightarrow } \cdots \stackrel{V_\lambda }{\longrightarrow } L^{q_{n_0}}(\Omega ) \stackrel{V_\lambda }{\longrightarrow } L^\infty (\Omega ) \end{aligned}$$

(that is, a bootstrap argument), we can show that $\sup \{\Vert V_\lambda (\eta (t,u))\Vert _\infty \,;\,t\in [0,\tau (u))\}<\infty$ holds by $\sup \{\Vert \eta (t,u)\Vert \,;\,t\in [0,\tau (u))\}<\infty$ because $V_\lambda$ is bounded from $L^{q_n}(\Omega )$ to $L^{q_{n+1}}(\Omega )$ and also from $L^{q_{n_0}}(\Omega )$ to $L^\infty (\Omega )$ due to Lemma 17 (iii) and (iv) , respectively. This boundedness leads to our claim.$\square$

3.2 Proof of Theorem 11

The next result follows from Lemma 17 (ii) and the argument in [24, Lemma 3.2.]. We omit the proof.

Lemma 19

If $u\in \pm \, P_B \setminus K$, then $\eta (t,u)\in \pm \,\mathrm{int}\,P_B$ for every $0<t<\tau (u)$.

The following result is well known (see [24, Lemma 2.3.]).

Lemma 20

Set

$$\begin{aligned} Q_\pm :=\{u\in X_B\setminus K\,;\, \eta (t,u)\in \pm \,\mathrm{int}\,P_B \ \mathrm{for\ some}\ t\in [0,\tau (u)) \,\}\cup (\pm \,\mathrm{int}\,P_B). \end{aligned}$$

(11)

Then, $Q_+$ and $Q_-$ are open subsets of $X_B$ and they are invariant for the descending flow $\eta$, that is, $\eta (t,u)\in Q_\pm$ for every $t\in [0,\tau (u))$ provided $u\in Q_\pm \setminus K$, respectively. In addition, $\partial Q_\pm$ are invariant closed subsets of $X_B$ for the descending flow $\eta$, where $\partial Q_\pm$ denotes the boundary of $Q_\pm$ in $X_B$.

An additional preliminary result is needed.

Lemma 21

If $J$ is coercive on $W_B$, then for every $\beta \in \mathbb R$, there exists an $R=R(\beta )>0$ such that $\Vert u\Vert \le R$ and $\Vert B_\lambda (u)\Vert \le R$ for every $u\in J^{-1}((-\infty ,\beta ])$.

Proof

By the coercivity of $J$, we have an $R_1>0$ satisfying $\Vert u\Vert \le R_1$ for every $u\in J^{-1}((-\infty ,\beta ])$. Let $u\in J^{-1}((-\infty ,\beta ])$ and $v=B_\lambda (u)$, namely $T_\lambda (v)=h(\cdot ,u)+\lambda \varphi _p(u)$ in $W_B^*$. By taking $v$ as a test function, we obtain

$$\begin{aligned}&\min \left\{ \frac{C_0}{p-1}, \lambda \right\} \Vert v\Vert ^p \le \int \limits _{\Omega }A(x,\nabla v)\nabla v\,\mathrm{d}x+\lambda \Vert v\Vert _p^p\\&\le (C+\lambda )\int \limits _{\Omega }|u|^{p-1}|v|\,\mathrm{d}x +C\Vert v\Vert _1 \le (C+\lambda )\Vert u\Vert _p^{p-1}\Vert v\Vert _p +C|\Omega |^{(p-1)/p}\Vert v\Vert _p\\&\le (C+\lambda )R_1^{p-1}\Vert v\Vert +C|\Omega |^{(p-1)/p}\Vert v\Vert \end{aligned}$$

[we use the Hölder’s inequality and Remark 6 (iii)], where $C$ is a positive constant as in (5). Because $p>1$, this yields the desired conclusion.$\square$

Proof of Theorem 11

First, we note that the boundary of $\pm P_B$ in $X_B$ includes no nontrivial critical points of $J$ by combining Lemma 15 and the fact that the critical points of $J$ are exactly the fixed points of $B_\lambda$. Due to the $(S)_+$ property of $V$ (see Proposition 8) and the compactness of $W_B\hookrightarrow L^p(\Omega )$, it is clear that $J$ satisfies the bounded Palais–Smale condition.

Choose a constant $\beta$ satisfying $\max _{t\in [0,1]} J(\gamma (t))<\beta <0$, where $\gamma$ is the continuous path in $(A2)$. Since it follows from Lemmas 19 and 20 that $\gamma (0)\in Q_+$, $\gamma (1)\in Q_-$ and $Q_\pm$ are open in $X_B$, there exist $0<t_+\le t_-<1$ such that $\gamma (t_+)\in \partial Q_+$ and $\gamma (t_-)\in \partial Q_-$ (it may happen that $t_+= t_-$ because it is not known whether $\partial Q_+\not =\partial Q_-$). Set $u_1:=\gamma (0)$, $u_2:=\gamma (1)$ and $u_3:=\gamma (t_+)$. Note that there exists an $R>0$ such that

$$\begin{aligned} \Vert \eta (t,u_i)\Vert \le R\quad \mathrm{and} \quad \Vert B_\lambda (\eta (t,u_i))\Vert \le R \quad \mathrm{for\ every}\ t\in [0,\tau (u_i)) \end{aligned}$$

(12)

by Lemma 21 and $\inf _{W_B}J \le J(\eta (t,u_i))\le \beta$ for every $t\in [0,\tau (u_i))$. Therefore, if $\tau (u_i)<\infty$ holds ($i=1,2,3$), then we have for every $0<t_1<t_2<\tau (u_i)<\infty$

$$\begin{aligned} \Vert \eta (t_1,u_i)-\eta (t_2,u_i)\Vert&\le \int \limits _{t_1}^{t_2}\Vert \eta (s,u_i)-V_\lambda (\eta (s,u_i))\Vert \,\mathrm{d}s \\&\le 2 \int \limits _{t_1}^{t_2}\Vert \eta (s,u_i)-B_\lambda (\eta (s,u_i))\Vert \,\mathrm{d}s \le 4R (t_2-t_1) \end{aligned}$$

by Lemma 17 (i) and (12). Hence, $\eta (t,u_i)$ converges to some $w_i$ in $W_B$ as $t\rightarrow \tau (u_i)-0$ provided $\tau (u_i)<\infty$. Moreover, according to Lemmas 18, 19 and 20, $w_i\in K$, $\eta (t,u_i)$ converges to $w_i$ in $X_B$ as $t\rightarrow \tau (u_i)-0$, $J(w_i)\le J(u_i)\le \beta <0$ ($i=1,2,3$) and $w_i\in \mathrm{int}\,P_B$ if $i=1$, $w_i\in -\mathrm{int}\,P_B$ if $i=2$ and $w_i\in \partial Q_+$ if $i=3$. Because of $\partial Q_+\cap (\pm P_B\setminus \{0\})=\emptyset$ (note that $\pm P_B\setminus \{0\} \subset Q_\pm$), our conclusion is shown when $\tau (u_i)<\infty$ for every $i=1,2,3$. Thus, we suppose that $\tau (u_i)=\infty$ for some $i\in \{1,2,3\}$. We claim that there exists a sequence $\{t_n\}\subset \mathbb R ^+$ such that

$$\begin{aligned} t_n\rightarrow +\infty \quad \mathrm{and}\quad J^\prime (\eta (t_n,u_i))\rightarrow 0 \quad \mathrm{in}\ W_B^* \quad \mathrm{as}\ n\rightarrow \infty . \end{aligned}$$

If our claim is shown, then it provides the existence of a Palais–Smale sequence of $J$ which is bounded because of (12). Thus, there exists $w_i\in W_B\cap K$ such that $\lim _{n\rightarrow \infty }\eta (t_n,u_i)=w_i$ in $W_B$ by choosing a subsequence if necessary. Furthermore, by the argument in Lemma 18 (iii) and (12), we see that $\{\eta (t,u_i)\,;\,t\ge 0\}$ is bounded in $C^{1,\alpha }_B(\overline{\Omega })$ for some $0<\alpha <1$. This yields that $\lim _{n\rightarrow \infty }\eta (t_n,u_i)=w_i$ in $X_B$ due to the compactness of $C^{1,\alpha }_B(\overline{\Omega })\hookrightarrow X_B$ and $\lim _{n\rightarrow \infty }\eta (t_n,u_i)=w_i$ in $W_B$. Therefore, there holds $w_i\in \mathrm{int}\,P_B$ if $i=1$, $w_i\in -\mathrm{int}\,P_B$ if $i=2$ and $w_i\in X_B\setminus (P_B\cup -P_B)$ if $i=3$.

Finally, we prove our claim. Note that there exists a sequence $\{t_n\}\subset \mathbb R ^+$ such that $t_n\rightarrow +\infty$ and $\frac{\mathrm{d}}{\mathrm{d}t} J(\eta (t_n,u_i))\rightarrow 0$ because $-\infty <\inf _{W_B}J \le J(\eta (t,u_i))\le \beta$ for every $t\ge 0$ and $J(\eta (t,u_i))$ is nondecreasing in $t$.

In the case of $1<p\le 2$, the following inequality follows from Lemma 16 (iii), Lemma 17 (i) and (12):

$$\begin{aligned} -\frac{\mathrm{d}}{\mathrm{d}t} J(\eta (t,u_i))&\ge \frac{d_1}{2}\Vert \eta (t,u_i)-B_\lambda (\eta (t,u_i))\Vert ^2 (\Vert \eta (t,u_i)\Vert +\Vert B_\lambda (\eta (t,u_i))\Vert )^{p-2}\\&\ge d_1 d_3^{-2/(p-1)}2^{p-3}R^{p-2} \Vert J^\prime (\eta (t,u_i))\Vert _{W_B^*}^{2/(p-1)} \end{aligned}$$

for every $t>0$. Similarly, in the case of $p>2$, we obtain

$$\begin{aligned} -\frac{\mathrm{d}}{\mathrm{d}t} J(\eta (t,u_i)) \ge 2^{2p-p^2-1} R^{2p-p^2} d_2 d_4^{-p}\Vert J^\prime (\eta (t,u_i))\Vert _{W_B^*}^{p} \end{aligned}$$

for every $t>0$. These inequalities applied for the sequence $\{t_n\}$ imply the proof of our claim.$\square$

4 Proof of Theorem 1 and Corollary 2

Throughout this section, we denote a super-solution and a sub-solution of $(P)_\mu$ in $(H1)$ by $u_\mu$ and $v_\mu$, respectively. We define

$$\begin{aligned} f_{[v_\mu ,u_\mu ]}(x,u):= \left\{ \begin{array}{ll} f(x,u_\mu (x)) \quad&\quad \mathrm{if}\quad \ u\ge u_\mu (x), \\ f(x,u) \quad&\quad \mathrm{if}\quad \ v_\mu (x)< u< u_\mu (x), \\ f(x,v_\mu (x)) \quad&\quad \mathrm{if}\quad \ u\le v_\mu (x), \end{array} \right. \end{aligned}$$

(13)

Moreover, we set

$$\begin{aligned} h_\mu (x,u):=\mu f_{[v_\mu ,u_\mu ]}(x,u) -p(u-u_\mu (x))_+^{p-1} +p(u-v_\mu (x))_-^{p-1}. \end{aligned}$$

(14)

Lemma 22

Assume $(H1)$ and $(H2)$. Then, for every $\mu >\mu _0$, there exists a $\lambda =\lambda (\mu )>0$ such that

$$\begin{aligned} h_\mu (x,u)u+\lambda |u|^p \ge 0\quad \text{ for} \text{ every}\ u\in \mathbb R ,\ \text{ a.e.}\ x\in \Omega . \end{aligned}$$

Proof

Because $f$ is bounded on each bounded set and by $(H2)$, there exists $\lambda _0>0$ such that

$$\begin{aligned} f(x,t)t +\lambda _0 |t|^p\ge 0 \quad \text{ for} \text{ every}\ |t|\le \max \{\Vert v_\mu \Vert _\infty , \Vert u_\mu \Vert _\infty \}, \ \text{ a.e.}\ x\in \Omega . \end{aligned}$$

(15)

Set $\lambda :=\mu \lambda _0 +p$. If $u_\mu (x)\ge t\ge v_\mu (x)$ holds, then $h_\mu (x,t)t+\lambda |t|^p =\mu (f(x,t)t+\lambda _0|t|^p) +p|t|^p\ge 0$ follows from (15). In the case of $t>u_\mu (x)(\ge 0)$, we have

$$\begin{aligned} h_\mu (x,t)t+\lambda |t|^p&= \mu t(f(x,u_\mu (x))+\lambda _0 u_\mu (x)^{p-1}) +\lambda _0 (t^{p-1}-u_\mu (x)^{p-1})\mu t\\&+p(t^{p-1} -(t-u_\mu (x))^{p-1})t\ge 0 \end{aligned}$$

since $s^{p-1}$ is nondecreasing on $\mathbb R ^+$ and (15). Similarly, we can show that $h_\mu (x,t)t+\lambda |t|^p \ge 0$ for $t<v_\mu (x)$.$\square$

Now, we introduce the functional $I_\mu$ on $W_B$ by

$$\begin{aligned} I_\mu (u)&:= \int \limits _{\Omega }G(x,\nabla u)\,\mathrm{d}x-\int \limits _{\Omega }\int \limits _0^{u(x)} h_\mu (x,t)\,\mathrm{d}t\mathrm{d}x\end{aligned}$$

(16)

$$\begin{aligned}&= \int \limits _{\Omega }G(x,\nabla u)\,\mathrm{d}x-\mu \int \limits _{\Omega }\int \limits _0^{u(x)} f_{[v_\mu ,u_\mu ]}(x,t)\,\mathrm{d}t\mathrm{d}x\nonumber \\&\qquad +\,\Vert (u-u_\mu )_+\Vert _p^p +\Vert (u-v_\mu )_-\Vert _p^p \end{aligned}$$

(17)

for $u\in W_B$ [see (2), (14) and (13) for the definitions of $G$, $h_\mu$ and $f_{[v_\mu ,u_\mu ]}$].

Because $f_{[v_\mu ,u_\mu ]}(\cdot ,u)\in L^\infty (\Omega )$ for every $u\in W_B$ by $u_\mu$, $v_\mu \in L^\infty (\Omega )$, we easily prove the following result due to the last two terms in (17) [that is, we use that $\Vert (u-w)_\pm \Vert _p /\Vert u_\pm \Vert _p\rightarrow 1$ as $\Vert u_\pm \Vert _p\rightarrow \infty$ for $w\in L^\infty (\Omega )$]. See Lemma 11 in [26] for the proof.

Lemma 23

Assume $(H1)$. Then, for every $\mu > \mu _0$, $I_\mu$ is coercive on $W_B$.

Moreover, we state the following important fact.

Lemma 24

Assume $(H1)$ and let $\mu > \mu _0$. If $u\in W_B$ is a critical point of $I_\mu$, then $u$ satisfies $v_\mu (x)\le u(x)\le u_\mu (x)$ for a.e. $x\in \Omega$. Therefore, $u$ is a solution of $(P)_\mu$ with $u\in [v_\mu ,u_\mu ]= \{w\in W_B\,;\, v_\mu \le w(x)\le u_\mu \ \mathrm{a.e.}\ x\in \Omega \}$.

Proof

This proof has been essentially done in [26, Lemma 14.]. For the readers’ convenience, we give it.

Let $u\in W_B$ be a critical point of $I_\mu$. Because $v_\mu$ and $u_\mu$ are a sub-solution and a super-solution of $(P)_\mu$, we have

$$\begin{aligned} \int \limits _{\Omega }A(x,\nabla v_\mu )\nabla w\,\mathrm{d}x&\le \mu \int \limits _{\Omega }f(x,v_\mu (x))w\,\mathrm{d}x =\mu \int \limits _{\Omega }f_{[v_\mu ,u_\mu ]}(x,v_\mu (x))w\,\mathrm{d}x\nonumber \\ \int \limits _{\Omega }A(x,\nabla u_\mu )\nabla w\,\mathrm{d}x&\ge \mu \int \limits _{\Omega }f(x,u_\mu (x))w\,\mathrm{d}x=\mu \int \limits _{\Omega }f_{[v_\mu ,u_\mu ]}(x,u_\mu (x))w\,\mathrm{d}x \end{aligned}$$

(18)

for every $w\in W_B$ with $w\ge 0$. Because of $(u-u_\mu )_+\in W_B$ (note that in the definition of a super-solution, we assume that $u_\mu \ge 0$ on $\partial \Omega$ in the Dirichlet problem), by taking $(u-u_\mu )_+$ as a test function in $I_\mu ^\prime (u)=0$ and (18), we have

$$\begin{aligned} 0&\ge \langle I_\mu ^\prime (u), (u-u_\mu )_+\rangle -\int \limits _{\Omega }A(x,\nabla u_\mu )\nabla (u-u_\mu )_+\,\mathrm{d}x\\&+\mu \int \limits _{\Omega }f_{[v_\mu ,u_\mu ]}(x,u_\mu (x))(u-u_\mu )_+\,\mathrm{d}x\\&= \int \limits _{u\ge u_\mu } (A(x,\nabla u)-A(x,\nabla u_\mu ))(\nabla u-\nabla u_\mu ) \, \mathrm{d}x +p\Vert (u-u_\mu )_+\Vert _p^p\ge 0 \end{aligned}$$

(note that $u_\mu \ge v_\mu$ and the map $A$ is strictly monotone in the second variable). This leads to $u(x)\le u_\mu (x)$ for a.e. $x\in \Omega$. Similarly, we obtain $u(x)\ge v_\mu (x)$ for a.e. $x\in \Omega$ by replacing $(u-u_\mu )_+$ and $u_\mu$ with $-(u-v_\mu )_-$ and $v_\mu$, respectively. Consequently, $u$ is a solution of $(P)_\mu$ with $v_\mu \le u\le u_\mu$ because of $h_\mu (x,u)=\mu f_{[v_\mu ,u_\mu ]}(x,u)=\mu f(x,u)$ (see Remark 12 for the boundary condition).$\square$

Lemma 25

Assume $(H1)$ and $(H3)$. Then, there exist $\mu _1>\mu _0$, $u_0\in P_B$ and $v_0\in -P_B$ such that

$$\begin{aligned} \max _{t\in [0,1]} I_\mu (t u_0+(1-t) v_0) <0 \quad \text{ for} \text{ every}\ \mu \ge \mu _1. \end{aligned}$$

Proof

Let $\Omega _1$ and $\Omega _2$ be open subsets as in $(H3)$. By taking new open subsets of $\Omega _1$ and $\Omega _2$ if necessary, we may assume that $\Omega _1\cap \Omega _2=\emptyset$.

Choose a nonnegative function $u_0\in C_0^\infty (\overline{\Omega }_1)$ and a nonpositive function $v_0\in C_0^\infty (\overline{\Omega }_2)$ such that $\Vert u_0\Vert _\infty < \min \{d_1,\delta _0\}$, $u_0>0$ somewhere, $\Vert v_0\Vert _\infty <\min \{d_2,\delta _0\}$ and $v_0<0$ somewhere, where $\delta _0$, $d_1$ and $d_2$ are positive constants as in $(H3)$.

Then, by the sign-condition for $f$ on $\Omega _1$ and $\Omega _2$ as in $(H3)$, we have $\int _{\Omega } F(x,tu_0)\,\mathrm{d}x>0$ and $\int _{\Omega } F(x,tv_0)\,\mathrm{d}x>0$ for every $0<t\le 1$, where $F(x,t):=\int _0^t f(x,s)\,\mathrm{d}s$. Define

$$\begin{aligned} d_+:=\min _{1\ge t\ge 1/2} \int \limits _{\Omega }F(x,tu_0)\,\mathrm{d}x>0, \quad d_-:=\min _{1\ge t\ge 1/2} \int \limits _{\Omega }F(x,tv_0)\,\mathrm{d}x>0, \end{aligned}$$

and choose a positive number $\mu _1$ satisfying $\mu _1>\mu _0$ and

$$\begin{aligned} \mu _1> \frac{C_1}{p(p-1)\min \{d_+,d_-\}} \left(\,\Vert \nabla u_0\Vert _p^p+\Vert \nabla v_0\Vert _p^p\right), \end{aligned}$$

(19)

where $C_1>0$ is the positive constant in $(A)$ (ii). Because $\mathrm{supp}\, u_0 \cap \mathrm{supp}\, v_0=\emptyset$, $0\le u_0\le d_1\le u_\mu$ in $\Omega _1$ and $0\ge v_0\ge -d_2\ge v_\mu$ in $\Omega _2$ hold, it is clear that

$$\begin{aligned} \int \limits _{\Omega }H_\mu (x,tu_0+(1-t)v_0)\,\mathrm{d}x =\mu \int \limits _{\Omega }F(x,tu_0)\,\mathrm{d}x +\mu \int \limits _{\Omega }F(x,(1-t)v_0)\,\mathrm{d}x \end{aligned}$$

for every $t\in [0,1]$, where $H_\mu (x,s):=\int _0^s h_\mu (x,\tau )\,\mathrm{d}\tau$. Therefore, we have

$$\begin{aligned} I_\mu (tu_0+(1-t)v_0)&= \int \limits _{\Omega _1} G(x,t\nabla u_0)\,\mathrm{d}x +\int \limits _{\Omega _2} G(x,(1-t)\nabla v_0)\,\mathrm{d}x\nonumber \\&\qquad -\mu \int \limits _{\Omega }F(x,tu_0)\,\mathrm{d}x-\mu \int \limits _{\Omega }F(x,(1-t)v_0)\,\mathrm{d}x\nonumber \\&\le \frac{C_1}{p(p-1)}\left(\,t^p\Vert \nabla u_0\Vert _p^p +(1-t)^p\Vert \nabla v_0\Vert _p^p\,\right)\nonumber \\&\qquad -\mu \int \limits _{\Omega }F(x,tu_0)\,\mathrm{d}x-\mu \int \limits _{\Omega }F(x,(1-t)v_0)\,\mathrm{d}x \end{aligned}$$

(20)

for every $t\in [0,1]$, where we use (3) in the last inequality. If $\mu \ge \mu _1$ and $0\le t\le 1/2$, then from (19), (20) and the definition of $d_-$,

$$\begin{aligned} I_\mu (tu_0+(1-t)v_0) \le \frac{C_1}{p(p-1)} \left(\,\Vert \nabla u_0\Vert _p^p+\Vert \nabla v_0\Vert _p^p\right) -\mu d_- <0 \end{aligned}$$

follows. Similarly, in the case of $\mu \ge \mu _1$ and $1\ge t\ge 1/2$, we easily obtain the inequality $I_\mu (tu_0+(1-t)v_0)\le C_1(\,\Vert \nabla u_0\Vert _p^p+\Vert \nabla v_0\Vert _p^p)/(p(p-1)) -\mu d_+<0$. Hence, our conclusion holds since $u_0\in P_B$ and $v_0\in -P_B$.$\square$

Lemma 26

Assume $(H1)$, $(H2)$ and $(H4)$. In addition, we suppose that $u_\mu \in \mathrm{int}\,P_B\cup \mathrm{int}\,P$ and $v_\mu \in -\mathrm{int}\,P_B\cup -\mathrm{int}\,P$ for every $\mu >\mu _0$. Then, there exist $\mu _1>\mu _0$, $u_0\in P_B$ and $v_0\in -P_B$ such that for every $\mu \ge \mu _1$ we can choose an $r_\mu >0$ satisfying

$$\begin{aligned} \max _{t\in [0,1]} I_\mu (t r_\mu u_0+(1-t) r_\mu v_0) <0. \end{aligned}$$

Proof

Let $m\in L^\infty (\Omega )$ and $\delta _1>0$ be as in $(H4)$. Then, since $|\{x\in \Omega \,;\,m(x)>0\}|>0$, we can take two open balls $B_1$ and $B_2$ such that $B_1\cap B_2 =\emptyset$ and $|\{x\in B_i\,;\,m(x)>0\}|>0$ ($i=1,2$) (refer to [15, Corollary 2, p 28] for the existence). It is well known that the first eigenvalue of the following weighted eigenvalue (Dirichlet) problem for the $p$-Laplacian on $B_i$ is positive and simple and that it has a positive eigenfunction belonging to $C_0^1(\overline{B}_i)$ (refer to [3] and [16, section 6.2]):

$$\begin{aligned} -\Delta _p u=\lambda m(x)|u|^{p-2}u \quad \mathrm{in}\ B_i,\quad u=0\quad \mathrm{on}\ \partial B_i,\quad (i=1,2), \end{aligned}$$

(21)

where $\Delta _p$ denotes the $p$-Laplacian. Therefore, since the above eigenvalue problem is $(p-1)$ homogeneous, for each $i=1,2$, there exist a positive solution $\psi _i\in C_0^1(\overline{B}_i)$ with $\Vert \psi _i\Vert _{C_0^1(\overline{B}_i)}=1$ of

$$\begin{aligned} -\Delta _p u=\lambda _1(m,B_i) m(x)|u|^{p-2}u \quad \mathrm{in}\ B_i,\quad u=0\quad \mathrm{on}\ \partial B_i, \end{aligned}$$

(22)

where $\lambda _1(m,B_i)>0$ denotes the first eigenvalue of (21). By taking $r^p\psi _i$ as a test function in (22), we have

$$\begin{aligned} r^p\lambda _1(m,B_i)\int \limits _{B_i} m\psi _i^{p}\,\mathrm{d}x =\int \limits _{B_i} |r\nabla \psi _i|^p\,\mathrm{d}x \ge \frac{p(p-1)}{C_1}\int \limits _{B_i} G(x,r\nabla \psi _i)\,\mathrm{d}x \end{aligned}$$

(23)

for every $r>0$, where we use (3) in the last inequality. Take $\mu _1>0$ satisfying $\mu _1>\mu _0$ and

$$\begin{aligned} \mu _1>\max \left\{ \,\frac{C_1\lambda _1(m,B_i)}{p(p-1)}\,;\,i=1,2\, \right\} . \end{aligned}$$

(24)

Since $u_\mu \in \mathrm{int}\,P_B\cup \mathrm{int}\,P$ and $v_\mu \in -\mathrm{int}\,P_B\cup -\mathrm{int}\,P$ for each $\mu >\mu _0$, there exists an $0<r_\mu <\delta _1$ such that $u_\mu -r_\mu \psi _i\in P_B \cup P$ and $v_\mu +r_\mu \psi _i\in -P_B\cup -P$ for $i=1,2$. As a result, it is easily shown that $v_\mu \le -r_\mu \psi _i< 0< r_\mu \psi _i\le u_\mu$ ($i=1,2$) and

$$\begin{aligned}&\int \limits _{\Omega }H_\mu (x, t r_\mu \psi _1-(1-t)r_\mu \psi _2)\,\mathrm{d}x\nonumber \\&\qquad =\mu \int \limits _{\Omega }F(x,t r_\mu \psi _1)\,\mathrm{d}x +\mu \int \limits _{\Omega }F(x, -(1-t)r_\mu \psi _2)\,\mathrm{d}x\nonumber \\&\qquad \ge \mu (t r_\mu )^p\int \limits _{\Omega }m \psi _1^p\,\mathrm{d}x +\mu (1-t)^pr_\mu ^p\int \limits _{\Omega }m \psi _2^p\,\mathrm{d}x \end{aligned}$$

(25)

for every $t\in [0,1]$ by $(H4)$ (note $\Vert r_\mu \psi _i\Vert _\infty <\delta _1$).

Therefore, for every $\mu \ge \mu _1$ and $t\in [0,1]$, we obtain

$$\begin{aligned} I_\mu (tr_\mu \psi _1 -(1-t)r_\mu \psi _2)&\le t^pr_\mu ^p\left(\,\frac{C_1}{p(p-1)}-\frac{\mu }{\lambda _1(m,B_1)}\,\right) \Vert \nabla \psi _1\Vert _p^p\\&\quad +(1-t)^pr_\mu ^p\left(\,\frac{C_1}{p(p-1)}-\frac{\mu }{\lambda _1(m,B_2)}\,\right) \Vert \nabla \psi _2\Vert _p^p <0 \end{aligned}$$

by (24), (25) and (23) with $tr_\mu \psi _1$ and $(1-t)r_\mu \psi _2$ in the place of $r\psi _i$.$\square$

Proof of Theorem 1

First, we note that it suffices to find critical points of $I_\mu$ in $\mathrm{int}\,P_B$, $-\mathrm{int}\,P_B$ and $X_B\setminus (P_B\cup -P_B)$ for sufficiently large $\mu$ according to Lemma 24. We already know that Lemmas 22, 25 or 26 imply $(A1)$ and $(A2)$ for $h_\mu$ and $I_\mu$ if $\mu > \max \{\mu _0,\mu _1\}$. Moreover, Lemma 23 ensures that $I_\mu$ is coercive on $W_B$ for every $\mu > \mu _0$. Therefore, by applying Theorem 11 to $I_\mu$ for each $\mu > \max \{\mu _0,\mu _1\}$, we obtain three critical points $w_{\mu ,1}\in \mathrm{int}\,P_B$, $w_{\mu ,2}\in -\mathrm{int}\,P_B$ and $w_{\mu ,3}\in X_B\setminus (P_B\cup -P_B)$ of $I_\mu$.$\square$

Proof of Corollary 2

(i) and (ii): By $(H5)$, we easily see that $T_+>0$ and $T_-<0$ are a super-solution and a sub-solution of $(P)_\mu$ for every $\mu >0$, respectively. Consequently, our conclusion follows from Theorem 1. (iii): Note that we are assuming $Bu=u=0$ (that is, Dirichlet boundary condition) in this case. Moreover, we also note that $(H2)$ follows from $(H4)$ (with $D_1=\Vert m\Vert _\infty$ and $\delta _0=\delta _1$).

According to Theorem 1, it is sufficient to obtain a super-solution in $\mathrm{int}\,P_B$ and a sub-solution in $-\mathrm{int}\,P_B$ of $(P)_\mu$ for each $\mu >0$. Here, we fix any $\mu >0$ and choose $\varepsilon >0$ satisfying

$$\begin{aligned} \varepsilon <\dfrac{\lambda _1 C_0}{\mu (p-1)}, \end{aligned}$$

(26)

where $\lambda _1$ denotes the first eigenvalue of $-\Delta _p$ in $\Omega$ under the Dirichlet boundary condition. By the condition $\mathrm{ess}\sup _{x\in \Omega }\limsup _{|u|\rightarrow \infty } f(x,u)/|u|^{p-2}u\le 0$, there exists an $R>0$ such that

$$\begin{aligned} \begin{array}{cl} f(x,u)\le \varepsilon u^{p-1}&\quad \mathrm{for}\ u\ge R, \ \mathrm{a.e.}\ x\in \Omega \\ \text{ and} \quad f(x,u)\ge -\varepsilon |u|^{p-1}&\quad \mathrm{for}\ u\le -R, \ \mathrm{a.e.}\ x\in \Omega . \end{array} \end{aligned}$$

(27)

We set $M_+:=\sup \{f(x,u)\,;\,x\in \Omega ,\ R\ge u\ge 0\}+1\ge 1$ and $M_-:=\inf \{f(x,u)\,;\,x\in \Omega ,\ -R\le u\le 0\}-1\le -1$ where the inequalities hold because $f$ is bounded on a bounded set and $f(x,0)=0$ for a.e. $x\in \Omega$. We define two functionals $I_\mu ^\pm$ on $W_B$ by

$$\begin{aligned} I_\mu ^\pm (u):=\int \limits _{\Omega }G(x,\nabla u)\,\mathrm{d}x - \mu M_\pm \int \limits _{\Omega }u\,\mathrm{d}x -\frac{\varepsilon \mu }{p}\int \limits _{\Omega }u_\pm ^{p}\,\mathrm{d}x \end{aligned}$$

for $u\in W_B$. Then, it is easily shown that $I_\mu ^\pm$ is coercive and bounded from below on $W_B$ by Poincaré’s inequality, (3) and (26). Furthermore, $\int _{\Omega } G(x,\nabla u)\,\mathrm{d}x$ is weakly lower semicontinuous (w.l.s.c.) on $W_B$ because $G$ is convex in the second variable (see Remark 6) and $\int _{\Omega } G(x,\nabla u)\,\mathrm{d}x$ is continuous on $W_B$ (see [25, Theorem 1.2.]). Thus, $I_\mu ^\pm$ is also w.l.s.c. on $W^{1,p}(\Omega )$ since the inclusion of $W_B$ into $L^p(\Omega )$ is compact. As a result, $I_\mu ^+$ and $I_\mu ^-$ have a global minimizer $u_{\mu }$ and $v_{\mu }$, respectively. Let us prove that $u_{\mu }\not =0$ and $v_{\mu }\not =0$. Indeed, by taking a positive eigenfunction $\varphi _1$ of $-\Delta _p$ corresponding to $\lambda _1$, we obtain

$$\begin{aligned} I_\mu ^+(u_{\mu })=\min _{W_B}I_\mu ^+ \le I_\mu ^+(t\varphi _1)\le \frac{t^pC_1}{p(p-1)}\Vert \nabla \varphi _1\Vert _p^p -t\mu M_+\Vert \varphi _1\Vert _1 -\frac{\varepsilon \mu t^p}{p}\Vert \varphi _1\Vert _p^p <0 \end{aligned}$$

for sufficiently small $t>0$ because of $p>1$ and $M_+>0$. Hence, $u_{\mu }\not =0$. Similarly, by considering $-t\varphi _1$, we have $v_{\mu }\not =0$.

We point out that $u_{\mu }\in X_B$ and $v_{\mu }\in X_B$ due to the regularity theorem in [23] because $u_{\mu }\in L^\infty (\Omega )$ and $v_{\mu }\in L^\infty (\Omega )$ by Moser’s iteration process (refer to Theorem C in [26] by noting that the nonlinearity satisfies the subcritical growth condition).

Next, we see that $u_{\mu }$ and $v_{\mu }$ are a super-solution and a sub-solution of $(P)_\mu$, respectively. Indeed, by the definition of $M_\pm$ and (27), we obtain

$$\begin{aligned} -\mathrm{div}\,A(x,\nabla u_{\mu })&=\mu M_+ +\mu \varepsilon (u_{\mu })_+^{p-1}\ge \mu f(x,u_{\mu })\\ -\mathrm{div}\,A(x,\nabla v_{\mu })&=\mu M_- -\mu \varepsilon (v_{\mu })_-^{p-1} \le \mu f(x,v_{\mu }) \end{aligned}$$

in $\Omega$. This shows our claim [note that $u_\mu$, $v_\mu \in X_B=C_0^1(\overline{\Omega })$].

Finally, we prove that $u_{\mu }\in \mathrm{int}\,P_B$ and $v_{\mu }\in -\mathrm{int}\, P_B$. In fact, by taking $-(u_{\mu })_-$ as a test function, we have

$$\begin{aligned} 0=\langle (I_\mu ^+)^\prime (u_{\mu }), -(u_{\mu })_-\rangle&= \int \limits _{\Omega }A(x,\nabla u_{\mu })(-\nabla (u_{\mu })_-)\,\mathrm{d}x +\mu M_+\int \limits _{\Omega }(u_{\mu })_-\,\mathrm{d}x\\&\ge \frac{C_0}{p-1}\Vert \nabla (u_{\mu })_-\Vert _p^p \ge \frac{\lambda _1 C_0}{p-1}\Vert (u_{\mu })_-\Vert _p^p \ge 0 \end{aligned}$$

by Remark 6 (iii), $M_+>0$ and Poincaré’s inequality. Thus, $(u_{\mu })_-(x)=0$ for every $x\in \Omega$, whence $u_{\mu }\ge 0$. Since $-\mathrm{div}\,A(x,\nabla u_{\mu }) =\mu M_+ +\mu \varepsilon u_{\mu }^{p-1}\ge 0$ in $\Omega$, we have $u_{\mu }(x)>0$ for every $x\in \Omega$ by Theorem B in [26] (note that $u_{\mu }\in X_B$ and $u_{\mu }\not \equiv 0$). In addition, due to the strong maximum principle (see Theorem A in [26]), we see that $\partial u_{\mu }(x)/\partial \nu <0$ for every $x\in \partial \Omega$. This implies $u_{\mu }\in \mathrm{int}\, P_B$.

Concerning $v_{\mu }$, by replacing $u_{\mu }$ with $-v_{\mu }$ in the above argument, we see that $-v_{\mu }\in \mathrm{int}\, P_B$ (note that $A$ is odd in the second variable).$\square$

5 Proofs in the special cases

First, in a similar way to $I_\mu$ as in Sect. 4, we define a functional $\widetilde{I}_\mu$ on $W_B$ as follows:

$$\begin{aligned} \widetilde{I}_\mu (u):=\int \limits _{\Omega }G(x,\nabla u)\,\mathrm{d}x -\int \limits _{\Omega }H(x,u,\mu )\,\mathrm{d}x \end{aligned}$$

for $u\in W_B$ and $\mu \in \fancyscript{M}$, where $H(x,u,\mu ):=\int _0^u h(x,t,\mu )\,\mathrm{d}x$ and

$$\begin{aligned} h(x,u,\mu )&:=\psi _1(\mu )f_{1,[v_\mu ,u_\mu ]}(x,u) +\psi _2(\mu )f_{2,[v_\mu ,u_\mu ]}(x,u) \\&\qquad -p(u-u_\mu (x))_+^{p-1}+p(u-v_\mu (x))_-^{p-1} \end{aligned}$$

with a super-solution $u_\mu$ and a sub-solution $v_\mu$ [see (13) for the definition of $f_{i,[v_\mu ,u_\mu ]}$].

Throughout this section, we denote a super-solution and a sub-solution of $(\widetilde{P})_\mu$ in $(\widetilde{H1})$ by $u_\mu$ and $v_\mu$, respectively.

By the same argument as in Sect. 4, we can prove the following two lemmas. Here, we omit the proofs.

Lemma 27

Assume $(\widetilde{H1})$. Then, for every $\mu \in \fancyscript{M}$, $\widetilde{I}_\mu$ is coercive on $W_B$.

Lemma 28

Assume $(\widetilde{H1})$. If $u\in W_B$ is a critical point of $\widetilde{I}_\mu$, then $u$ satisfies $v_\mu (x)\le u(x)\le u_\mu (x)$ for a.e. $x\in \Omega$. Therefore, $u$ is a solution of $(\widetilde{P})_\mu$ within the order interval $[v_\mu ,u_\mu ]$.

Lemma 29

Assume $(\widetilde{H1})$, $(\widetilde{H2})$, $(\widetilde{H3})$ and $(\widetilde{H4})$. Then, for every $\mu \in \fancyscript{M}$, there exists a $\lambda =\lambda (\mu )>0$ such that

$$\begin{aligned} h(x,u,\mu )u+\lambda |u|^p\ge 0 \quad \text{ for} \text{ every}\ u\in \mathbb R ,\ \text{ a.e.}\ x\in \Omega . \end{aligned}$$

Proof

Fix any $\mu \in \fancyscript{M}$. Because $f_1$ is bounded on each bounded set and by $(\widetilde{H2})$, there exists $\lambda _1>0$ such that

$$\begin{aligned} f_1(x,t)t+\lambda _1 |t|^p\ge 0 \quad \mathrm{for\ every}\ |t|\le \max \{\Vert v_\mu \Vert _\infty ,\Vert u_\mu \Vert _\infty \}, \ \mathrm{a.e.}\ x\in \Omega . \end{aligned}$$

Moreover, there exists $\lambda _2>0$ such that

$$\begin{aligned} f_2(x,t)t+\lambda _2 |t|^p\ge 0 \quad \mathrm{for\ every}\ |t|\le \max \{\Vert v_\mu \Vert _\infty ,\Vert u_\mu \Vert _\infty \}, \ \mathrm{a.e.}\ x\in \Omega . \end{aligned}$$

since $f_2(x,t)t$ is positive for sufficiently small $|t|>0$ by $(\widetilde{H3})$ and $f_2$ is also bounded on each bounded set. Set $\lambda =\psi _1(\mu )\lambda _1+\psi _2(\mu )\lambda _2+p>0$. By the same argument as in Lemma 22, we can reach our conclusion.$\square$

Lemma 30

Assume $(\widetilde{H1})$, $(\widetilde{H2})$, $(\widetilde{H3})$ and $(\widetilde{H4})$. Then, for every $\mu \in \fancyscript{M}$, there exist $w_\mu ^1\in P_B$ and $w_\mu ^2\in -P_B$ such that

$$\begin{aligned} \max _{t\in [0,1]} \widetilde{I}_\mu \left(tw_\mu ^1 +(1-t)w_\mu ^2\right)<0. \end{aligned}$$

Proof

First, we choose smooth functions $u_0\not \equiv 0$ and $v_0\not \equiv 0$ satisfying $\mathrm{supp}\, u_0\cap \mathrm{supp}\,v_0=\emptyset$ and $u_0\ge 0\ge v_0$ in $\overline{\Omega }$. Fix any $\mu \in \fancyscript{M}$. Since $u_\mu \in \mathrm{int}\,P_B \cup \mathrm{int}\,P$ and $v_\mu \in -\mathrm{int}\,P_B\cup -\mathrm{int}\,P$ [see $(\widetilde{H1})$], there exists an $r_\mu >0$ such that $u_\mu \pm ru_0\in P_B\cup P$ and $v_\mu \pm r v_0\in -P_B\cup -P$ for every $0<r\le r_\mu$. Because of $\mathrm{supp}\, u_0\cap \mathrm{supp}\,v_0=\emptyset$, this implies that $u_\mu \ge tru_0\ge 0 \ge (1-t)rv_0\ge v_\mu$ and

$$\begin{aligned}&\int \limits _{\Omega }H(x,tr u_0+(1-t)r v_0,\mu )\,\mathrm{d}x\nonumber \\&\quad =\sum _{i=1}^2 \psi _i(\mu )\int \limits _{\Omega }F_i(x,tr u_0)\,\mathrm{d}x +\sum _{i=1}^2 \psi _i(\mu )\int \limits _{\Omega }F_i(x,(1-t)r v_0)\,\mathrm{d}x \end{aligned}$$

(28)

for every $0<r\le r_\mu$ and $t\in [0,1]$, where $F_i(x,s):=\int _0^s f_i(x,\tau )\,\mathrm{d}\tau$ ($i=1,2$). By the hypothesis $(\widetilde{H3})$, there exist $\delta _1>0$ and $\rho _1>0$ such that

$$\begin{aligned} f_2(x,u)u\ge \beta \rho _1|u|^\beta \quad \mathrm{for\ every}\ |u|<\delta _1. \end{aligned}$$

Thus, we have

$$\begin{aligned} \int \limits _{\Omega }F_2(x,w)\,\mathrm{d}x \ge \rho _1\Vert u\Vert _\beta ^\beta \quad \mathrm{for\ every}\ w\in L^\infty (\Omega )\ \mathrm{with}\ \Vert w\Vert _\infty <\delta _1. \end{aligned}$$

(29)

Moreover, it follows from $(\widetilde{H2})$ that

$$\begin{aligned} \int \limits _{\Omega }F_1(x,w)\,\mathrm{d}x \ge -\frac{D_1}{p}\Vert w\Vert _p^p \quad \mathrm{for\ every}\ w\in L^\infty (\Omega )\ \mathrm{with}\ \Vert w\Vert _\infty <\delta _0, \end{aligned}$$

(30)

where $D_1$ and $\delta _0$ are the positive constants in $(\widetilde{H2})$. Here, to simplify the notation, we set

$$\begin{aligned} \Phi (r,w):=C_1r^{p-\beta }\Vert \nabla w\Vert _p^p +D_1(p-1)r^{p-\beta }\psi _1(\mu )\Vert w\Vert _p^p -\rho _1p(p-1)\psi _2(\mu )\Vert w\Vert _\beta ^\beta \end{aligned}$$

for $w\in W_B$. Then, by $p>\beta >1$, $\psi _2(\mu )>0$, $\Vert u_0\Vert _\beta >0$ and $\Vert v_0\Vert _\beta >0$, we can choose an $r>0$ such that

$$\begin{aligned} r<\min \left\{ r_\mu , \frac{\delta _0}{\Vert u_0\Vert _\infty }, \frac{\delta _0}{\Vert v_0\Vert _\infty }, \frac{\delta _1}{\Vert u_0\Vert _\infty },\frac{\delta _1}{\Vert v_0\Vert _\infty }\right\} , \Phi (r,u_0)<0\ \mathrm{and}\ \Phi (r,v_0)<0. \end{aligned}$$

(31)

Therefore, for every $t\in [0,1]$ and such $r>0$, we obtain

$$\begin{aligned}&\widetilde{I}_\mu (tru_0+(1-t)rv_0)\\&\quad =\int \limits _{\Omega }G(x,tr\nabla u_0)\,\mathrm{d}x +\int \limits _{\Omega }G(x,(1-t)r\nabla v_0)\,\mathrm{d}x\\&\qquad -\sum _{i=1}^2 \psi _i(\mu )\int \limits _{\Omega }F_i(x,tr u_0)\,\mathrm{d}x -\sum _{i=1}^2 \psi _i(\mu )\int \limits _{\Omega }F_i(x,(1-t)r v_0)\,\mathrm{d}x\\&\quad \le \frac{C_1\left((tr)^p\Vert \nabla u_0\Vert _p^p + (1-t)^pr^p\Vert \nabla v_0\Vert _p^p\right)}{p(p-1)} +\frac{D_1\psi _1(\mu )\left(\Vert tru_0\Vert _p^p +\Vert (1-t)rv_0\Vert _p^p\right)}{p}\\&\qquad -\rho _1\psi _2(\mu ) \left(\Vert tru_0\Vert _\beta ^\beta +\Vert (1-t)rv_0\Vert _\beta ^\beta \right)\\&\quad \le \frac{(tr)^\beta }{p(p-1)} \Phi (r,u_0) + \frac{(1-t)^\beta r^\beta }{p(p-1)} \Phi (r,v_0)<0 \end{aligned}$$

by (3), (28), (29), (30) and (31).$\square$

Proof of Theorem 3

Fix any $\mu \in \fancyscript{M}$. It follows from Lemma 27 that $\widetilde{I}_\mu$ is coercive on $W_B$. According to Lemma 29 and Lemma 30, $h(x,u,\mu )$ satisfies $(A1)$ and $(A2)$ holds for $J=\widetilde{I}_\mu$. Thus, by applying Theorem 11, $\widetilde{I}_\mu$ has three critical points $w_1\in \mathrm{int}\,P_B$, $w_2\in -\mathrm{int}\,P_B$ and $w_3\in X_B\setminus (P_B\cup -P_B)$. Moreover, Lemma 28 guarantees that they are solutions of $(\widetilde{P})_\mu$ with $w_i\in [u_\mu ,v_\mu ]$ ($i=1,2,3$). The proof is complete.$\square$

Proof of Corollary 5

• (i) Since the constant function $T(\mu )_+> 0$ is a super-solution and $T(\mu )_-< 0$ is a sub-solution of $(\widetilde{P})_\mu$, our conclusion follows from Theorem 3.

• (ii) First, we recall that in this case we assume the Dirichlet boundary condition. According to Theorem 3, it suffices to prove that for every $\mu \in \fancyscript{M}$, there exists a super-solution $u_\mu \in \mathrm{int}\,P_B$ and a sub-solution $v_\mu \in -\mathrm{int}\,P_B$ of $(\widetilde{P})_\mu$. Fix $\mu \in \fancyscript{M}$. Because $\mathrm{ess}\sup _{x\in \Omega }\,\limsup _{|u|\rightarrow \infty }\tilde{f}(x,u,\mu )/|u|^{p-2}u<C_0\lambda _1/(p-1)$ there exist $\varepsilon >0$ and $R>0$ such that

$$\begin{aligned}&\tilde{f}(x,u,\mu )\le \frac{C_0(\lambda _1-\varepsilon )}{p-1} u^{p-1} \quad \mathrm{for}\ u\ge R,\ \mathrm{a.e.}\ x\in \Omega ,\\ \mathrm{and} \quad&\tilde{f}(x,u,\mu )\ge -\frac{C_0(\lambda _1-\varepsilon )}{p-1} |u|^{p-1} \quad \mathrm{for}\ u\le -R,\ \mathrm{a.e.}\ x\in \Omega . \end{aligned}$$

Set $M_+:=\sup \{\tilde{f}(x,u,\mu )\,;\,0\le u\le R,\ x\in \Omega \}+1>0$ and $M_-:=\inf \{\tilde{f}(x,u,\mu )\,;\,0\ge u\ge -R,\ x\in \Omega \}-1<0$. Define the functionals $\widetilde{I}_\mu ^\pm$ on $W_0^{1,p}(\Omega )$ by

$$\begin{aligned} \widetilde{I}_\mu ^\pm (u):=\int \limits _{\Omega }G(x,\nabla u)\,\mathrm{d}x -M_\pm \int \limits _{\Omega }u\,\mathrm{d}x -\frac{C_0(\lambda _1-\varepsilon )}{p(p-1)}\int \limits _{\Omega }u^p_\pm \,\mathrm{d}x \end{aligned}$$

for $u\in W_0^{1,p}(\Omega )$. Due to the same argument as in Corollary 2, these functionals have a global minimizer with $\min \widetilde{I}_\mu ^\pm <0$. Moreover, this guarantees the existence of a super-solution in $\mathrm{int}\,P_B$ and a sub-solution in $-\mathrm{int}\,P_B$ (refer to Corollary 2 for details). Thus, our conclusion holds.$\square$

Appendix

For readers’ convenience, we give the proof of Proposition 10. We remark that we do not need inequality (35) in the following proof under the Dirichlet boundary condition. However, it is necessary for our approach because we consider the Neumann problem too.

Proof of Proposition 10

Let $u\in L^r(\Omega )$ and $v=T_\lambda ^{-1}(u)(\in W_B)$ with $r>N/p$. Consider $\bar{p}^*$ such that $\bar{p}^*:=p^*$ if $N>p$ and $\bar{p}^*>pr/(r-1)$ if $N=p$.

Set $k=\Vert u\Vert _r^{1/(p-1)}$ and for $\varphi =v_+$ or $v_-$, we put $\varphi _M(x):=\min \,\{\varphi (x), M\}$ for $M>0$ to simplify the notation. For $q\ge 1$, we define

$$\begin{aligned} H(z):= {\left\{ \begin{array}{ll} 0&\mathrm{if}\ z<k,\\ z^q -k^q&\mathrm{if}\ k\le z. \end{array}\right.} \end{aligned}$$

Let $w:=\varphi +k$ and $w_M:=\varphi _M +k$. Define

$$\begin{aligned} K(w_M):=\int \limits _k^{w_M} H^\prime (s)^p\,\mathrm{d}s. \end{aligned}$$

Note that $H^\prime (s)\ge 0$ for every $s\not =0$ and that if $\varphi \in W_0^{1,p}(\Omega )$, then $w_M=k$ on $\partial \Omega$ in the sense of the trace operator, and hence $K(w_M)\in W_B$.

First, we consider $\varphi =v_+$. By taking $K(w_M)$ as a test function in $T_\lambda (v)=u$, we obtain

$$\begin{aligned}&\int \limits _{\Omega }A(x,\nabla v)\nabla K(w_M)\,\mathrm{d}x +\lambda \int \limits _{\Omega }|v|^{p-2}v K(w_M)\,\mathrm{d}x \end{aligned}$$

(32)

$$\begin{aligned}&= \int \limits _{\Omega }u K(w_M)\,\mathrm{d}x \le \int \limits _{\Omega }|u|\,\left(\int \limits _k^{w_M} H^\prime (s)^p\,\mathrm{d}s \right)\, \mathrm{d}x \nonumber \\&\le \int \limits _{\Omega }|u| w_M H^\prime (w_M)^p\,\mathrm{d}x \le k^{-p+1}\int \limits _{\Omega }|u|w_M^p H^\prime (w_M)^p\,\mathrm{d}x\nonumber \\&\le k^{-p+1}\Vert u\Vert _r\Vert w_M H^\prime (w_M)\Vert _{pr/(r-1)}^p =\Vert w_M H^\prime (w_M)\Vert _{pr/(r-1)}^p, \end{aligned}$$

(33)

by Hölder’s inequality and $k=\Vert u\Vert _r^{1/(p-1)}$, where we use $w_M/k\ge 1$ and $H^\prime (s)\le H^\prime (w_M)$ for every $s\le w_M$. On the other hand, the inequality

$$\begin{aligned} \int \limits _{\Omega }A(x,\nabla v)\nabla K(w_M)\,\mathrm{d}x&= \int \limits _{\Omega }A(x,\nabla v)\nabla \varphi _M H^\prime (w_M)^p\,\mathrm{d}x \nonumber \\&\ge \frac{C_0}{p-1} \int \limits _{\Omega }|\nabla \varphi _M|^p H^\prime (w_M)^p\,\mathrm{d}x =\frac{C_0}{p-1}\Vert \nabla H(w_M)\Vert _p^p\nonumber \\ \end{aligned}$$

(34)

follows from Remark 6 (iii), $w_M=\varphi _M+k$ and $v_+=\varphi$. Concerning the second term in the left-hand side of (32), we have

$$\begin{aligned}&\int \limits _{\Omega }|v|^{p-2}v K(w_M)\,\mathrm{d}x =\int \limits _{v\ge 0} v^{p-1} \int \limits _k^{w_M} H^\prime (s)^p\,\mathrm{d}s\mathrm{d}x \nonumber \\&\ge \int \limits _{v\ge 0}\int \limits _k^{w_M} H^\prime (s) \left((w_M-k)H^\prime (s)\right)^{p-1}\,\mathrm{d}s\mathrm{d}x \nonumber \\&=\int \limits _{v\ge 0}\int \limits _{2k}^{w_M+k} H^\prime (s-k) \big (\,(w_M-k)H^\prime (s-k)\,\big )^{p-1}\,\mathrm{d}s\mathrm{d}x \nonumber \\&\ge \int \limits _{v\ge 0}\int \limits _{2k}^{w_M+k} H^\prime (s-k) (H(s-k))^{p-1}\,\mathrm{d}s\mathrm{d}x =\frac{1}{p}\int \limits _{\Omega }H(w_M)^p\,\mathrm{d}x \end{aligned}$$

(35)

by $(v_+=)\varphi \ge \varphi _M=w_M-k$ and $(w_M-k)H^\prime (s-k)\ge q(s-k)^q-kq(s-k)^{q-1}\ge H(s-k)$ for $2k\le s\le w_M+k$. Because $W_B\hookrightarrow L^{\bar{p}^*}(\Omega )$ is continuous, according to (33), (34) and (35), there exists a $C_*>0$ such that

$$\begin{aligned} C_*\min \left\{ \frac{C_0}{p-1},\frac{\lambda }{p}\right\} \Vert H(w_M)\Vert _{\bar{p}^*}^p \le \Vert w_M H^\prime (w_M)\Vert _{pr/(r-1)}^p=q^p\Vert w_M\Vert _{pqr/(r-1)}^{pq}. \end{aligned}$$

(36)

Similarly, by taking $-K(w_M)$ as a test function in $T_\lambda (v)=u$ for the case of $\varphi =v_-$, we have the same inequality as above. Consequently, we have

$$\begin{aligned} C_*^\prime \Vert w_M\Vert _{\bar{p}^*q}^q&\le q\Vert w_M\Vert _{pqr/(r-1)}^q +C_*^\prime k^q|\Omega |^{1/\bar{p}^*}\nonumber \\&\le q(1+C_*^\prime |\Omega |^{1/\bar{p}^*+(1-r)/(pr)})\Vert w_M\Vert _{pqr/(r-1)}^q \end{aligned}$$

for every $q\ge 1$, where $C^\prime _*=C_*^{1/p}\min \{C_0/(p-1),\lambda /p\}^{1/p}$ and we use Holder’s inequality, $k^q=\Vert k\Vert _q^q/|\Omega |$ and $k\le w_M$ in the last inequality. This yields

$$\begin{aligned} \Vert w_M\Vert _{\bar{p}^*q}\le (qD_*)^{1/q}\Vert w_M\Vert _{\alpha q} \end{aligned}$$

(37)

for every $q\ge 1$ with $\alpha :=pr/(r-1)$, where $D_*>0$ is a positive constant independent of $M$, $k$, $u$, $v$. Note that $\alpha <\bar{p}^*$ by our assumption $r>N/p$ and the definition of $\bar{p}^*$. At this point, we define a sequence $\{q_n\}$ by $q_0:=\bar{p}^*/\alpha (>1)$ and $q_{n+1}:=q_n\bar{p}^*/\alpha =(\bar{p}^*/\alpha )^{n+2}$. Then, by taking $M\rightarrow \infty$ in (37), we see that if $w\in L^{\alpha q_n}(\Omega )$, then $w\in L^{\alpha q_{n+1}}(\Omega )$ and it satisfies $\Vert w\Vert _{\alpha q_{n+1}}\le (q_nD_*)^{1/q_n}\Vert w\Vert _{\alpha q_n}$. This leads to $\Vert w\Vert _\infty \le C \Vert w\Vert _{\bar{p}^*}$ by a standard argument, where $C$ is a positive constant independent of $w$ and $n$. Therefore, we have

$$\begin{aligned} \Vert \varphi \Vert _\infty \le C\Vert \varphi \Vert _{\bar{p}^*} +k(1+C|\Omega |^{1/\bar{p}^*})= C\Vert \varphi \Vert _{\bar{p}^*} +\Vert u\Vert _r^{1/(p-1)}(1+C|\Omega |^{1/\bar{p}^*}) \end{aligned}$$

(38)

(note $w=\varphi +k$ and $k=\Vert u\Vert _r^{1/(p-1)}$). On the other hand, by taking $\pm v_\pm$ as test function in $T_\lambda (v)=u$,

$$\begin{aligned} C_*\min \left\{ \frac{C_0}{p-1},\lambda \right\} \Vert \varphi \Vert _{\bar{p}^*}^p&\le \min \left\{ \frac{C_0}{p-1},\lambda \right\} \Vert \varphi \Vert ^p \le \langle T_\lambda (v), \pm v_\pm \rangle \nonumber \\&\le \Vert u\Vert _r\Vert \varphi \Vert _{r/(r-1)}\le \Vert u\Vert _r\Vert \varphi \Vert _{\bar{p}^*} |\Omega |^\beta \end{aligned}$$

(39)

(note that $\varphi$ denotes $v_+$ or $v_-$), where we use Remark 6 (iii) in the second inequality and Hölder’s inequality setting $\beta =((r-1)\bar{p}^*-r)/(r-1)\bar{p}^*$. From (38) and (39), our conclusion follows.$\square$