Multiple solutions for superlinear double phase Neumann problems

Abstract

We study a double phase Neumann problem with a superlinear reaction which need not satisfy the Ambrosetti-Rabinowitz condition. Using the Nehari manifold method, we show that the problem has at least three nontrivial bounded ground state solutions, all with sign information (positive, negative and nodal).

1 Introduction

Let $\Omega \subseteq \mathbb {R}^N$ be a bounded domain with a Lipschitz boundary $\partial \Omega$. In this paper we study the following double phase Neumann problem

$$\begin{aligned} \left\{ \begin{array}{ll} - \Delta _p^a u(z) - \Delta _q u(z) + \xi (z) |u(z)|^{q - 2} u(z) = f(z,u(z)) \text { in } \Omega ,\\ \displaystyle \frac{\partial u}{\partial n} = 0 \text { on } \partial \Omega ,\ 1< q< p < N. \end{array} \right\} \end{aligned}$$

(1)

Given $a \in L^\infty (\Omega ) \setminus \{0\}$ with $a(z) \ge 0$ for a.a. $z \in \Omega$ and $1< r < + \infty$, we denote by $\Delta _r^a$ the weighted r-Laplace differential operator defined by

$$\begin{aligned} \Delta _r^a u = \mathrm{div\,} \left( a(z) |Du|^{r - 2} Du\right) . \end{aligned}$$

If $a \equiv 1$, then we have the usual r-Laplacian denoted by $\Delta _r$. The differential operator in problem (1) is the sum of two such operators with different exponents. Hence the differential operator is not homogeneous. There is also a potential term $\xi (z) |u|^{q - 2} u$ with $\xi \in L^\infty (\Omega ) \setminus \{0\}$, $\xi (z) \ge 0$ for a.a. $z \in \Omega$. The differential operator of problem (1) is related to the so-called “double phase" integral functional defined by

$$\begin{aligned} \int _\Omega \left( a(z) |Du|^p + |Du|^q\right) dz. \end{aligned}$$

The integrand of this functional is

$$\begin{aligned} \theta (z,x) = a(z) x^p + x^q \text { for all } z \in \Omega , \text { all } x \ge 0. \end{aligned}$$

We do not assume that the weight $a(\cdot )$ is bounded away from zero (that is, we do not have that ${\mathrm{ess\,inf}}_{\Omega } a > 0$). Therefore the integrand $\theta (z,x)$ exhibits unbalanced growth, namely

$$\begin{aligned} x^q \le \theta (z,x) \le c_0 \left( x^q + x^p \right) \text { for a.a. } z \in \Omega , \text { all } x \ge 0, \text { some } c_0 > 0. \end{aligned}$$

Such functionals, wre first considered by Marcellini [14, 15] and Zhikov [27, 28], in the context of problems of the calculus of variations and of nonlinear elasticity theory. Recently the interest for such problems was revived and some important interior regularity results were obtained for local minimizers of such functionals. We refer to the works of Baroni et al. [1], Mingione and Rădulescu [16], and Ragusa and Tachikawa [24]. However, until now there is no global regularity theory (that is, regularity results up to the boundary) for the solutions of such problems. This fact limits the tools available for the study of such equations and makes double phase problems more difficult to deal with. In fact, the unbalanced growth of $\theta (z,\cdot )$ implies that the usual Sobolev spaces do not suffice for the analysis of the problem and we have to use the general abstract setting of Musielak-Orlicz-Sobolev spaces.

In the reaction of problem (1), we have a Carathéodory function f(z, x) (that is, for all $x \in \mathbb {R}$ the mapping $z \mapsto f(z,x)$ is measurable and for a.a. $z \in \Omega$ the function $x \mapsto f(z,x)$ is continuous). We assume that $f(z,\cdot )$ exhibits $(p-1)$-superlinear growth as $x \rightarrow \pm \infty$ but without satisfying the usual in such cases Ambrosetti-Rabinowitz condition (the AR-condition for short). The absence of a global regularity theory for such problems requires a different approach for double phase equations. In problems with balanced growth ((p, q)-equations), for which a powerful global regularity theory exists (see Lieberman [10]), the main tools are truncation and comparison techniques, critical point theory and Morse theory (critical groups). We refer to the works of Liu and Papageorgiou [12], Papageorgiou and Rădulescu [17, 18], Papageorgiou et al. [21], Papageorgiou and Zhang [23]. Here instead, we use the Nehari manifold method as this was developed by Brown and Wu [2], Brown and Zhang [3], Szulkin and Weth [25] and Willem [26].

We show that problem (1) has at least three nontrivial bounded solutions, all with sign information (positive, negative and nodal (sign changing)). Recently there have been some multiplicity results for double phase problems. We refer to the works of Gasiński and Papageorgiou [5], Ge et al. [8], Liu and Dai [11], Papageorgiou et al. [22] (Dirichlet problems) and Gasiński and Winkert [7], Papageorgiou, Rădulescu and Repovš [20] (Neumann and Robin problems). Closer to our work here are the papers of Gasiński and Papageorgiou [5] and Liu and Dai [11], which deal with superlinear Dirichlet problems, employing more restrictive conditions on the reaction and using a different approach.

2 Mathematical background and hypotheses

As we already mentioned in the introduction, in order to accommodate the unbalanced growth of $\theta (z,\cdot )$, we have to use Musielak-Orlicz-Sobolev spaces. A comprehensive presentation of the theory of these spaces can be found in the recent book of Harjulehto and Hästö [9].

We make the following hypotheses on the weight $a(\cdot )$, the potential function $\xi (\cdot )$ and exponents q, p.

$\mathrm{H_0}$: $a \in C^{0,1}(\overline{\Omega })$ (that is, $a: \overline{\Omega } \mapsto \mathbb {R}$ is Lipschitz continuous), $a(z) > 0$ for all $z \in \Omega$, $\xi \in L^\infty (\Omega ) \setminus \{0\}$, $\xi (z) \ge 0$ for a.a. $z \in \Omega$, $1< q< p < q^*$ $\left( q^* = \displaystyle \frac{Nq}{N - q}\right)$, $p < N$.

Remark 1

In contrast to previous works on Dirichlet double phase problems,  we do not require that $\displaystyle \frac{p}{q} < 1 + \displaystyle \frac{1}{N}.$ This condition implies that $p < q^*,$ but the converse is not true. For Dirichlet problems this condition guarantees that the Poincaré inequality is valid for the corresponding Musielak-Orlicz-Sobolev space (see Harjulehto & Hästö [9, pp. 100, 138]).

Let $\mathrm{M}(\Omega )$ be the space of all Lebesgue measurable functions $u : \Omega \mapsto \mathbb {R}$. As usual we identify two measurable functions which differ only on a Lebesgue-null subset of $\Omega$. Then the Musielak-Orlicz space $L^\theta (\Omega )$ is defined by

$$\begin{aligned} L^\theta (\Omega ) = \left\{ u \in \mathrm{M}(\Omega ) : \rho _\theta (u) < + \infty \right\} , \end{aligned}$$

with $\rho _\theta (\cdot )$ being the modular function defined by

$$\begin{aligned} \rho _\theta (u) = \int _\Omega \theta (z,u(z)) dz = \int _\Omega \left( a(z) |u(z)|^p + |u(z)|^q\right) dz. \end{aligned}$$

We equip $L^\theta (\Omega )$ with the so-called “Luxemburg norm"

$$\begin{aligned} \Vert u\Vert _\theta = \inf \left\{ \lambda > 0 : \rho _\theta \left( \displaystyle \frac{ u}{\lambda }\right) \le 1\right\} . \end{aligned}$$

Then $L^\theta (\Omega )$ becomes a Banach space which is separable and reflexive (in fact, uniformly convex).

Using $L^\theta (\Omega )$ we can define the corresponding Musielak-Orlicz-Sobolev space $W^{1,\theta }(\Omega )$ as follows

$$\begin{aligned} W^{1,\theta }(\Omega ) = \left\{ u \in L^\theta (\Omega ) : |Du| \in L^\theta (\Omega )\right\} . \end{aligned}$$

Here Du denotes the weak gradient of $u(\cdot )$. We equip $W^{1,\theta }(\Omega )$ with the following norm

$$\begin{aligned} \Vert u\Vert = \Vert u\Vert _\theta + \Vert Du\Vert _\theta \text { for all } u \in W^{1,\theta }(\Omega ), \end{aligned}$$

where $\Vert Du\Vert _\theta = \Vert |Du|\Vert _\theta$. This space is also a Banach space which is separable and reflexive (in fact, uniformly convex).

We have the following embeddings for these spaces (see Gasiński and Winkert [7]).

Proposition 1

If hypotheses $\mathrm{H_0}$ hold,  then the following embeddings are true : 

• (a) $L^\theta (\Omega ) \hookrightarrow L^r(\Omega )$ and $W^{1,\theta }(\Omega ) \hookrightarrow W^{1,r}(\Omega )$ continuously for all $1 \le r \le q;$

• (b) $W^{1,\theta }(\Omega ) \hookrightarrow L^r(\Omega )$ continuously if $1 \le r \le q$ and compactly if $1 \le r < q;$

• (c) $L^p(\Omega ) \hookrightarrow L^\theta (\Omega )$ continuously.

There is a close relation between the norm $\Vert \cdot \Vert _\theta$ and the modular function $\rho _\theta (\cdot )$.

Proposition 2

If hypotheses $\mathrm{H_0}$ hold,  then

• (a) if $u \in L^\theta (\Omega )\!\setminus \!\{0\},$ then $\Vert u\Vert _\theta = \mu$ $\Longleftrightarrow$ $\rho _\theta \left( \displaystyle \frac{u}{\mu }\right) = 1;$

• (b) $\Vert u\Vert _\theta < 1$ (resp.$\ = 1,\ > 1)$ $\Longleftrightarrow$ $\rho _\theta (u) < 1$ (resp.$\ = 1,\ > 1);$

• (c) $\Vert u\Vert _\theta < 1$ $\Longrightarrow$ $\Vert u\Vert _\theta ^p \le \rho _\theta (u) \le \Vert u\Vert _\theta ^q;$

• (d) $\Vert u\Vert _\theta > 1$ $\Longrightarrow$ $\Vert u\Vert _\theta ^q \le \rho _\theta (u) \le \Vert u\Vert _\theta ^p;$

• (e) $\Vert u_n\Vert _\theta \rightarrow 0$ $\Longleftrightarrow$ $\rho _\theta (u_n) \rightarrow 0;$

• (f) $\Vert u_n\Vert _\theta \rightarrow + \infty$ $\Longleftrightarrow$ $\rho _\theta (u_n) \rightarrow + \infty .$

Let $\langle \cdot ,\cdot \rangle$ denote the duality brackets for the pair $\langle W^{1,\theta }(\Omega )^*, W^{1,\theta }(\Omega ) \rangle$ and let $V : W^{1,\theta }(\Omega ) \mapsto W^{1,\theta }(\Omega )^*$ be the nonlinear operator defined by

$$\begin{aligned} \langle V(u), h \rangle = \int _\Omega a(z)|Du|^{p - 2}(Du,Dh)_{\mathbb {R}^N} dz + \int _\Omega |Du|^{q - 2}(Du,Dh)_{\mathbb {R}^N} dz \end{aligned}$$

for all u, $h \in W^{1,\theta }(\Omega )$.

The next proposition summarizes the main properties of this nonlinear operator (see Liu and Dai [11]).

Proposition 3

If hypotheses $\mathrm{H_0}$ hold,  then the operator $V : W^{1,\theta }(\Omega ) \mapsto W^{1,\theta }(\Omega )^*$ is bounded (that is,  it maps bounded sets to bounded sets),  continuous,  monotone and of type $(S)_+,$ namely it has the following property : 

$$\begin{aligned} ``u_n \xrightarrow {w} u \text { in } W^{1,\theta }(\Omega ), \ \limsup \limits _{n \rightarrow \infty }\langle V(u_n),u_n-u \rangle \le 0 \end{aligned}$$

imply that

$$\begin{aligned} u_n \rightarrow u \text { in } W^{1,\theta }(\Omega ).'' \end{aligned}$$

For every $u \in W^{1,\theta }(\Omega )$, we set

$$\begin{aligned} |u| = \Vert Du\Vert _\theta + \left( \int _\Omega \xi (z)|u|^q dz\right) ^{1/q}. \end{aligned}$$

Evidently, this is a norm on $W^{1,\theta }(\Omega )$.

Proposition 4

If hypotheses $\mathrm{H_0}$ hold,  then $\Vert \cdot \Vert$ and $|\cdot |$ are equivalent norms on $W^{1,\theta }(\Omega ).$

Proof

Since $\xi \in L^\infty (\Omega )$ (see hypotheses $\mathrm{H_0}$), we have

$$\begin{aligned} \int _\Omega \xi (z) |u|^q dz \le \Vert \xi \Vert _\infty \Vert u\Vert _q^q \le c_1 \Vert u\Vert _q^q \end{aligned}$$

for some $c_1 > 0$ (see Proposition 1).

Therefore

$$\begin{aligned} |u| = \Vert Du\Vert _\theta + c_1^{1/q}\Vert u\Vert _\theta \le c_2 \Vert u\Vert \end{aligned}$$

(2)

for some $c_2 > 0$, all $u \in W^{1,\theta }(\Omega )$.

Claim: There exists $c_3 > 0$ such that $c_3 \Vert u\Vert \le |u|$ for all $u \in W^{1,\theta }(\Omega )$.

Arguing by contradiction, suppose we can find $\left\{ u_n\right\} _{n \in \mathbb {N}} \subseteq W^{1,\theta }(\Omega )$ such that

$$\begin{aligned} |u_n| < \displaystyle \frac{1}{n} \Vert u_n\Vert \text { for all } n \in \mathbb {N}. \end{aligned}$$

Let $v_n = \displaystyle \frac{u_n}{\Vert u_n\Vert }$ ($n \in \mathbb {N}$). Then we have

$$\begin{aligned} \Vert v_n\Vert = 1 \text { and } \Vert Dv_n\Vert _\theta + \left( \int _\Omega \xi (z) |v_n|^q dz \right) ^{1/q} < \displaystyle \frac{1}{n} \text { for all } n \in \mathbb {N}. \end{aligned}$$

(3)

By passing to a subsequence if necessary, we may assume that

$$\begin{aligned} v_n \xrightarrow w v \text { in } W^{1,\theta }(\Omega ) \text { and } v_n \rightarrow v \text { in } L^q(\Omega )\ (\text {see Proposition~1}). \end{aligned}$$

(4)

Then from (3) and (4), we have

$$\begin{aligned}&\Vert Dv\Vert _\theta + \left( \int _\Omega \xi (z) |v|^q dz \right) ^{1/q} \le 0,\nonumber \\ \Rightarrow \ \&\Vert Dv\Vert _\theta + \left( \int _\Omega \xi (z) |v|^q dz \right) ^{1/q} = 0, \nonumber \\ \Rightarrow \ \&\Vert Dv\Vert _\theta = 0 \text { and so } v = \eta \in \mathbb {R}. \end{aligned}$$

(5)

From (5) we have

$$\begin{aligned} |\eta |^q \int _\Omega \xi (z) dz = 0. \end{aligned}$$

Since $\displaystyle \int _\Omega \xi (z) dz > 0$ (see hypotheses $\mathrm{H_0}$), we have that $\eta = 0$ and so $v =0$. From (3) we have

$$\begin{aligned}&\Vert Dv_n\Vert _\theta \rightarrow 0,\\ \Rightarrow \ \&v_n \rightarrow 0 \text { in } W^{1,\theta }(\Omega ), \end{aligned}$$

which contradicts the fact that $\Vert v_n\Vert = 1$ for all $n \in \mathbb {N}$ (see (3)). Finally, (2) and the Claim imply that

$$\begin{aligned} \Vert \cdot \Vert \text { and } |\cdot | \text { are equivalent norms on } W^{1,\theta }(\Omega ). \end{aligned}$$

This proof is now complete. $\square$

In a similar fashion we can show also the following result. In what follows, $\Vert \cdot \Vert _{1,q}$ denotes the norm of the Sobolev space $W^{1,q}(\Omega )$.

Proposition 5

If $\xi \in L^\infty (\Omega )\!\setminus \!\{0\}$ and $\xi (z) \ge 0$ for a.a. $z \in \Omega ,$ then $\Vert \cdot \Vert _{1,q}$ and $[u] = \Vert Du\Vert _q + \left( \displaystyle \int _\Omega \xi (z) |u|^q dz \right) ^{1/q}$ are equivalent norms on $W^{1,q}(\Omega ).$

Given $u \in W^{1,\theta }(\Omega )$, we define

$$\begin{aligned} u^+(z) = \max \left\{ u(z),0\right\} ,\ u^-(z) = \max \left\{ - u(z),\ 0 \right\} \text { for all }z \in \Omega . \end{aligned}$$

We know that

$$\begin{aligned} u^\pm \in W^{1,\theta }(\Omega ),\ u = u^+ - u^-,\ |u| = u^+ + u^-. \end{aligned}$$

If X is a Banach space and $\varphi \in C^1(X,\mathbb {R})$, then

$$\begin{aligned} K_\varphi = \left\{ u \in X : \varphi '(u) = 0\right\} \ (\text {the critical set of } \varphi ). \end{aligned}$$

Also, if $c \in \mathbb {R}$, then

$$\begin{aligned} \varphi ^c = \left\{ u \in X : \varphi (u) \le c\right\} . \end{aligned}$$

Now we introduce the hypotheses on the reaction f(z, x).

$\mathrm{H_1}$: $f : \Omega \times \mathbb {R}\mapsto \mathbb {R}$ is a Carathéodory function such that $f(z,0) = 0$ for a.a. $z \in \Omega$ and

• (i) $|f(z,x)| \le \hat{a}(z) \left( 1 + |x|^{r -1}\right)$ for a.a. $z \in \Omega$, all $x \in \mathbb {R}$ with $\hat{a} \in L^\infty (\Omega )$ and $p< r < q^*$;

• (ii) if $F(z,x) = \displaystyle \int _0^x f(z,s) ds$, then $\lim \nolimits _{x \rightarrow \pm \infty } \displaystyle \frac{F(z,x)}{|x|^p} = + \infty$ uniformly for a.a. $z \in \Omega$;

• (iii) for a.a. $z \in \Omega$, the quotient function $x \mapsto \displaystyle \frac{f(z,x)}{|x|^{p - 1}}$ is increasing on $\mathbb {R}\!\setminus \!\left\{ 0\right\}$;

• (iv) $\lim \nolimits _{x \rightarrow 0} \displaystyle \frac{f(z,x)}{|x|^{q- 2} x} = 0$ uniformly for a.a. $z\in \Omega$.

Remark 2

For $s \ge 0$ and $x \ne 0,$ we have

$$\begin{aligned}&\frac{1 - s^p}{p} f(z,x) x + F(z,sx) - F(z,x) \nonumber \\ =\ \&\int _s^1 f(z,x) x t^{p -1} dt - \int _s^1 \frac{d}{dt} F(z,tx) dt\nonumber \\ =\ \&\int _s^1 f(z,x) x t^{p -1} dt - \int _s^1 f(z,tx) x dt \ (\text {by the chain rule})\nonumber \\ =\ \&\int _s^1 \left( \frac{f(z,x)}{|x|^{p - 1}} - \frac{f(z,tx)}{t^{p - 1} |x|^{p - 1}}\right) |x|^{p -1} x t^{p - 1} dt \nonumber \\ \ge \ \&0\ (\text {see hypothesis } \mathrm{H_1(iii)} ),\nonumber \\ \Rightarrow \ \&F(z,x) - F(z,sx) \le \frac{1 - s^p}{p} f(z,x) x \nonumber \\&\text { for a.a. } z \in \Omega ,\text { all } x \in \mathbb {R}, \text { all } s \ge 0. \end{aligned}$$

(6)

If in (6) we choose $s = 0$, then

$$\begin{aligned}&p F(z,x) \le f(z,x) x \text { for a.a. } z \in \Omega , \text { all } x \in \mathbb {R}, \\ \Rightarrow \ \&\lim \limits _{x \rightarrow \pm \infty } \frac{f(z,x)}{|x|^{p - 2} x} = + \infty \text { uniformly for a.a. } z \in \Omega \ (\text {see hypothesis }\mathrm{H_1(ii)}).\nonumber \end{aligned}$$

(7)

So, we see that $f(z,\cdot )$ is $(p - 1)$-superlinear as $x \rightarrow \pm \infty$. However this superlinearity property is not formulated using the AR-condition which is common in superlinear problems, see Liu & Dai [11], hypothesis $(f_4)$. Also, in hypothesis $\mathrm{H_1(iii)}$ we do not require that the quotient function $x \mapsto \displaystyle \frac{f(z,x)}{|x|^{p - 1}}$ is strictly increasing on $\mathbb {R}\setminus \{0\}$ as it is common in the literature, see Liu & Dai [11], hypothesis $(f_6)$. Compared to the other superlinear Dirichlet double phase work of Gasiński & Papageorgiou [5], we do not require $f(z,\cdot )$ to be locally Lipschitz and replace hypothesis $\mathrm{H}(f)\mathrm{(iii)}$ of [5] by hypothesis $\mathrm{H_1(iii)}$. In case $f(z,\cdot )$ is strictly differentiable, then hypothesis $\mathrm{H}(f)\mathrm{(iii)}$ of [5] has the form $(p - 1) f(z,x) x \le f_x'(z,x) x^2$ for a.a. $z \in \Omega$, all $x \in \mathbb {R}$ (see Clarkc [4, p. 33]), which of course implies the monotonicity of the quotient function. Note that hypothesis $\mathrm{H_1(iii)}$ is equivalent to saying that for a.a. $z \in \Omega$, the mapping $x \mapsto \displaystyle \frac{f(z,x)x}{|x|^p}$ is increasing in $|x| \ne 0$.

First we will study problem (1) under the following stronger set of hypotheses.

$\mathrm{H_1'}$: $f : \Omega \times \mathbb {R}\mapsto \mathbb {R}$ is a Carathéodory function such that $f(z,0) = 0$ for a.a. $z \in \Omega$, hypotheses $\mathrm{H_1'(i)}$, (ii), (iv) are the same as the corresponding hypotheses $\mathrm{H_1(i)}$, (ii), (iv) and

(iii) for a.a. $z \in \Omega$, the quotient function $x \mapsto \displaystyle \frac{f(z,x)}{|x|^{p - 1}}$ is strictly increasing on $\mathbb {R}\!\setminus \!\{0\}$.

Example 1

The following function satisfies hypotheses $\mathrm{H_1}$ but not $\mathrm{H_1'}.$ For the sake of simplicity we drop the z-dependence.

$$\begin{aligned} f(x) = {\left\{ \begin{array}{ll} \ |x|^{p - 2} x &{}\text { if } |x| \le 1,\\ \ |x|^{p - 2} x \left( \ln |x| + 1\right) &{}\text { if } 1 < |x|. \end{array}\right. } \end{aligned}$$

This function does not satisfy the AR-condition and so it does not fit in the framework of Liu & Dai [11]. Also it does not fit the framework of Gasiński & Papageorgiou [5] since it does not satisfy hypothesis $\mathrm{H}(f)\mathrm{(iii)}$ of [5].

In what follows, $\rho _a(\cdot )$ denotes the modular function defined by

$$\begin{aligned} \rho _a(Du) = \int _\Omega a(z) |Du|^p dz \text { for all } u \in W^{1,\theta }(\Omega ). \end{aligned}$$

Let $\varphi : W^{1,\theta }(\Omega ) \mapsto \mathbb {R}$ be the energy (Euler) functional for problem (1) defined by

$$\begin{aligned} \varphi (u) = \frac{1}{p} \rho _a(Du) + \frac{1}{q} \Vert Du\Vert _q^q + \frac{1}{q} \int _\Omega \xi (z) |u|^q dz - \int _\Omega F(z,u) dz \end{aligned}$$

for all $u \in W^{1,\theta }(\Omega )$.

We have that $\varphi \in C^1(W^{1,\theta }(\Omega ))$ and the Nehari manifold N of $\varphi (\cdot )$, is defined by

$$\begin{aligned} \mathrm{N} = \left\{ u \in W^{1,\theta }(\Omega ) : \langle \varphi '(u),u \rangle = 0,\ u \ne 0\right\} . \end{aligned}$$

Evidently, every weak solution of problem (1) is in the Nehari manifold. Since we want to produce constant sign solution for problem (1), we introduce the $C^1$-functionals $\varphi _\pm : W^{1,\theta }(\Omega ) \mapsto \mathbb {R}$ defined by

$$\begin{aligned} \varphi _\pm (u) = \frac{1}{p} \rho _a(Du) + \frac{1}{q} \Vert Du\Vert _q^q + \frac{1}{q} \int _\Omega \xi (z) |u|^q dz - \int _\Omega F(z,\pm u^\pm ) dz. \end{aligned}$$

We associate with these functionals, the following submanifolds of N

$$\begin{aligned}&\mathrm{N}_+ = \left\{ u \in W^{1,\theta }(\Omega ) : \langle \varphi _+'(u), u \rangle = 0,\ u \ge 0,\ u \ne 0 \right\} ,\\&\mathrm{N}_- = \left\{ u \in W^{1,\theta }(\Omega ) : \langle \varphi _-'(u), u \rangle = 0,\ u \le 0,\ u \ne 0 \right\} . \end{aligned}$$

Finally, in order to produce a nodal (sign changing) solution of problem (1), we introduce the set

$$\begin{aligned} \mathrm{N}_0 = \left\{ u \in W^{1,\theta }(\Omega ) : \langle \varphi '(u),u^+ \rangle = \langle \varphi '(u),u^-\rangle = 0,\ u^\pm \ne 0\right\} . \end{aligned}$$

Note that

$$\begin{aligned} \mathrm{N}_+,\ \mathrm{N}_-,\ \mathrm{N}_0 \subseteq \mathrm{N}. \end{aligned}$$

(8)

For the study of N and of its submanifolds, we introduce the fibering function. So, let $u \in W^{1,\theta }(\Omega )\!\setminus \!\{0\}$ and consider the function

$$\begin{aligned} k_u(t) = \varphi (tu) \text { for all } t \ge 0. \end{aligned}$$

This is the fibering function for u. Note that

$$\begin{aligned} t u \in \mathrm{N} \text { if and only if } k_u'(t) = 0. \end{aligned}$$

3 Constant sign solutions under strict monotonicity

In this section, working with the functionals $\varphi _\pm$ and the submanifolds $\mathrm{N}_\pm$, we will produce two constant sign (positive and negative) solutions for problem (1) under $\mathrm{H_1'}$.

The Nehari manifold N is much smaller than the space $W^{1,\theta }(\Omega )$. So, the energy functional $\varphi (\cdot )$ restricted on N may exhibit properties which fail globally. In our case note that on account of hypothesis $\mathrm{H_1(ii)}$, $\varphi (\cdot )$ is unbounded below. However, as we show in the next proposition, $\varphi |_\mathrm{N}$ is coercive, thus bounded below.

Proposition 6

If hypotheses $\mathrm{H_0},$ $\mathrm{H_1}$ hold,  then $\varphi |_\mathrm{N}$ is coercive.

Proof

Arguing by contradiction, suppose we can find $\left\{ u_n\right\} _{n \in \mathbb {N}} \subseteq W^{1,\theta }(\Omega )$ such that

$$\begin{aligned} u_n \in \mathrm{N} \text { for all } n \in \mathbb {N},\ \Vert u_n\Vert \rightarrow + \infty ,\ \varphi (u_n) \le c_4 \text { for some } c_4 > 0, \text { all } n \in \mathbb {N}. \end{aligned}$$

(9)

Since $u_n \in \mathrm{N}$, we have

$$\begin{aligned} \rho _a(Du_n) + \Vert Du_n\Vert _q^q + \int _\Omega \xi (z) |u|^q dz&= \int _\Omega f(z,u_n) u_n dz, \nonumber \\ \Rightarrow \ \ \frac{1}{p} \rho _a(Du_n) + \frac{1}{p} \Vert Du_n\Vert _q^q + \frac{1}{p} \int _\Omega \xi (z) |u_n|^q dz&= \frac{1}{p} \int _\Omega f(z,u_n) u_n dz\nonumber \\&\ge \int _\Omega F(z,u_n) dz \nonumber \\&\ \ \text { for all } n \in \mathbb {N}\ (\text {see (7)}). \end{aligned}$$

(10)

From (9) and (10), we obtain

$$\begin{aligned}&\left( \frac{1}{q} - \frac{1}{p}\right) \left( \Vert Du_n\Vert _q^q + \int _\Omega \xi (z) |u_n|^q dz\right) \le c_4 \text { for all } n \in \mathbb {N}, \end{aligned}$$

(11)

$$\begin{aligned} \Rightarrow \ \&\left\{ u_n\right\} _{n \in \mathbb {N}} \subseteq W^{1,q}(\Omega ) \text { is bounded (see Proposition 5)},\nonumber \\ \Rightarrow \ \&\left\{ u_n\right\} _{n \in \mathbb {N}} \subseteq L^r(\Omega ) \text { is relatively compact (see Proposition 1)}. \end{aligned}$$

(12)

From (9), (11) and (12) it follows that

$$\begin{aligned}&\frac{1}{p} \rho _a(Du_n) \le c_5 \text { for some } c_5 > 0, \text { all } n \in \mathbb {N}, \end{aligned}$$

(13)

$$\begin{aligned} \Rightarrow \ \&\rho _\theta (Du_n) + \int _\Omega \xi (z) |u_n|^q dz \le c_6 \text { for some } c_6 > 0, \text { all } n \in \mathbb {N}\ (\text {see (11)}, (13)),\nonumber \\ \Rightarrow \ \&\left\{ u_n\right\} _{n \in \mathbb {N}} \subseteq W^{1,\theta }(\Omega ) \text { is bounded (see Proposition 4)}. \end{aligned}$$

(14)

Comparing (9) and (14), we have a contradiction. This proves that $\varphi | _\mathrm{N}$ is coercive. $\square$

Clearly we have

$$\begin{aligned} \varphi _+ |_\mathrm{N_+} = \varphi | _\mathrm{N_+}, \ \varphi _- |_\mathrm{N_-} = \varphi | _\mathrm{N_-}. \end{aligned}$$

Since $\mathrm{N_+}$, ${}\mathrm N_- \subseteq \mathrm{N}$ (see (8)), from Proposition 6 we infer the following corollary.

Corollary 7

If hypotheses $\mathrm{H_0}$, $\mathrm{H_1}$ hold,  then $\varphi _+ |_\mathrm{N_+}$ and $\varphi _- |_\mathrm{N_-}$ are both coercive.

Next we show that $\mathrm{N} \ne \emptyset$. Now we will start using the stronger hypotheses $\mathrm{H_1'}$.

Proposition 8

If hypotheses $\mathrm{H_0},$ $\mathrm{H_1'}$ hold and $u \in W^{1,\theta }(\Omega )\!\setminus \!\{0\},$ then there exists a unique $t_u > 0$ such that

$$\begin{aligned} t_u u \in \mathrm{N}. \end{aligned}$$

Proof

We introduce the function $\beta _u : \mathbb {R}_+ \mapsto \mathbb {R}$ defined by

$$\begin{aligned} \beta _u(t) = t^p \rho _a(Du) + t^q \left( \Vert Du\Vert _q^q + \int _\Omega \xi (z) |u|^q dz \right) - \int _\Omega f(z,tu) (tu) dz \end{aligned}$$

(15)

for all $t \ge 0$.

We see that

$$\begin{aligned} \beta _u(t) = t k_u'(t) \text { for all } t \ge 0. \end{aligned}$$

(16)

Hypothesis $\mathrm{H_1'(iii)}$ implies that if $x \ne 0$ and $t \in (0,1)$, then

$$\begin{aligned}&\frac{f(z,tx)(tx)}{t^p |x|^p}< \frac{f(z,x) x}{|x|^p} \text { for a.a. } z \in \Omega ,\nonumber \\ \Rightarrow \ \&f(z,tx) (tx) < t^p f(z,x) x \text { for a.a. } z \in \Omega , \text { all } x \ne 0, \text { all } t \in (0,1). \end{aligned}$$

(17)

We use (17) in (15) and obtain

$$\begin{aligned} \beta _u(t)&\ge t^p \rho _a(Du) + t^q \left( \Vert Du\Vert _q^q + \int _\Omega \xi (z) |u|^q dz\right) - t^p \int _\Omega f(z,u) u dz\\&\ge t^q \frac{1}{2^{q -1}} [u]^q - t^p \int _\Omega f(z,u) u dz. \end{aligned}$$

Since $q < p$, we see that

$$\begin{aligned} \beta _u(t) > 0 \text { for all } t \in (0,1) \text { small}. \end{aligned}$$

(18)

Also we have

$$\begin{aligned} \frac{\beta _u(t)}{t^p}&= \rho _a(Du) + \frac{1}{t^{p - q}} \left( \Vert Du\Vert _q^q + \int _\Omega \xi (z) |u|^q dz\right) - \int _\Omega \frac{f(z,tu) (tu)}{t^p} dz\nonumber \\&\le \rho _a(Du) + \frac{2}{t^{p -q}} [u]^q - \int _\Omega \frac{p F(z,tu)}{t^p} dz \ (\text {see (7)}),\nonumber \\ \Rightarrow \ \&\lim \limits _{t \rightarrow + \infty } \frac{\beta _u(t)}{t^p} = - \infty \ (\text {see hypothesis } \mathrm{H_1'(ii)}),\nonumber \\ \Rightarrow \ \&\beta _u(t) < 0 \text { for } t > 0 \text { large}. \end{aligned}$$

(19)

Then relations (18), (19) and Bolzano’s theorem, imply that there exists $t_u > 0$ such that

$$\begin{aligned}&\beta _u(t_u) = 0,\\ \Rightarrow \ \&t_u k_u'(t_u) = 0\ (\text {see (16)}),\\ \Rightarrow \ \&k_u'(t_u) = 0,\\ \Rightarrow \ \&t_u u \in \mathrm{N}. \end{aligned}$$

Finally, note that

$$\begin{aligned} \beta _u(t) = 0 \Longleftrightarrow \rho _u(Du) = \int _\Omega \frac{f(z,tu)(tu)}{t^p} dz - \frac{1}{t^{p - q}} \left( \Vert Du\Vert _q^q + \int _\Omega \xi (z) |u|^q dz\right) . \end{aligned}$$

In the last equality the right-hand side is strictly increasing in $t > 0$ (see hypothesis $\mathrm{H_1'(ii)}$) and so it follows that $t_u > 0$ is unique. $\square$

In a similar fashion, working with $\varphi _+$ and $\varphi _-$, we show the nonemptiness of the submanifolds $\mathrm{N_+}$ and $\mathrm{N_-}$, respectively. Recall that $\mathrm{N_\pm } \subseteq \mathrm{N}$ (see (8)).

Proposition 9

If hypotheses $\mathrm{H_0},$ $\mathrm{H_1'}$ hold and $u \in W^{1,\theta }(\Omega ) \setminus \{0\}$ with $u \ge 0$ (resp. $u \le 0$),  then there exists a unique $t_u^+ > 0$ (resp. $t_u^- > 0$) such that

$$\begin{aligned} t_u^+ u \in \mathrm{N_+}\ (\text {resp}. \ t_u^- u \in \mathrm{N_-}). \end{aligned}$$

The next proposition presents a useful connection between the fibering function and the elements of the Nehari manifold.

Proposition 10

If hypotheses $\mathrm{H_0},$ $\mathrm{H_1'}$ hold and $u \in \mathrm{N},$ then $\varphi (tu) \le \varphi (u)$ for all $t \ge 0.$

Proof

From (16) we have

$$\begin{aligned}&\beta _u(1) = k_u'(1),\\ \Rightarrow \ \&\beta _u(1) = 0 \ (\text {since } u \in \mathrm{N}). \end{aligned}$$

Note that

$$\begin{aligned}&\frac{k_u(t)}{t^p} = \frac{1}{p} \rho _a(Du) + \frac{1}{q t^{p - q}} \left( \Vert Du\Vert _q^q + \int _\Omega \xi (z) |u|^q dz\right) - \int _\Omega \frac{F(z,tu)}{t^p} dz,\nonumber \\ \Rightarrow \ \&\frac{k_u(t)}{t^p} \rightarrow - \infty \text { as } t \rightarrow + \infty \ (\text {see hypothesis } { H_1'(ii)}),\nonumber \\ \Rightarrow \ \&k_u(t) < 0 \text { for all } t >0 \text { large}. \end{aligned}$$

(20)

On the other hand, on account of hypotheses $\mathrm{H_1'(i)}$, (iv) given $\varepsilon > 0$, we can find $c_7 = c_7(\varepsilon ) > 0$ such that

$$\begin{aligned} F(z,x) \le \frac{\varepsilon }{q} |x|^q + c_7 |x|^r \text { for a.a. } z \in \Omega , \text { all } x \in \mathbb {R}. \end{aligned}$$

(21)

Then using (21) we have

$$\begin{aligned} k_u(t) \ge \frac{t^p}{p} \rho _a(Du) + \frac{t^q}{q} \left( \frac{1}{2^{q - 1}} - \varepsilon c_8\right) [u]^q - t^r c_7 \Vert u\Vert _r^r \end{aligned}$$

for some $c_8 > 0$ (see Proposition 5).

Choosing $\varepsilon \in \left( 0, \displaystyle \frac{1}{2^{q - 1} c_8}\right)$, we obtain

$$\begin{aligned}&k_u(t) \ge c_9 t^p - c_{10} t^r \text { for some } c_9,\ c_{10}> 0, \text { all } t \in (0,1),\nonumber \\ \Rightarrow \ \&k_u(t) > 0 \text { for all } t \in (0,1) \text { small } \ (\text {recall that } p < r). \end{aligned}$$

(22)

From (21), (22) it follows that there exists $\hat{t}_u > 0$ local maximizer of the fibering function. Then

$$\begin{aligned}&k_u'(\hat{t}_u) = 0,\\ \Rightarrow \ \&\hat{t}_u u \in \mathrm{N},\\ \Rightarrow \ \&\hat{t}_u = 1 \text { and it is a global maximizer of } k_u(\cdot )\\&(\text {recall that } u \in \mathrm{N} \text { and see Proposition 8}),\\ \Rightarrow \ \&\varphi (t u) \le \varphi (u) \text { for all } t \ge 0. \end{aligned}$$

This proof is now complete. $\square$

The same result is also true for $\varphi _\pm$.

Proposition 11

If hypotheses $\mathrm{H_0},$ $\mathrm{H_1'}$ hold and $u \in \mathrm{N_+}$ (resp. $u \in \mathrm{N_-}$),  then $\varphi _+(tu) \le \varphi _+(u)$ (resp. $\varphi _-(tu) \le \varphi _-(u)$) for all $t \ge 0.$

In the next proposition we show that the elements of the Nehari manifold are in norm bounded away from zero.

Proposition 12

If hypotheses $\mathrm{H_0},$ $\mathrm{H_1'}$ hold,  then there exists $\mu _0 > 0$ such that

$$\begin{aligned} 0 < \mu _0 \le \Vert u\Vert _r,\ \Vert u\Vert \text { for all } u \in \mathrm{N}. \end{aligned}$$

Proof

From hypotheses $\mathrm{H_1'(i)}$, (iv), we see that given $\varepsilon > 0$, we can find $c_{11} = c_{11}(\varepsilon ) > 0$ such that

$$\begin{aligned} f(z,x) x \le \varepsilon |x|^q + c_{11} |x|^r \text { for a.a. } z \in \Omega , \text { all } x \in \mathbb {R}. \end{aligned}$$

(23)

Let $u \in \mathrm{N}$ and we can always assume that $\Vert u\Vert$, $|u| \le 1$. Then

$$\begin{aligned}&\rho _a(Du) + \left( \Vert Du\Vert _q^q + \int _\Omega \xi (z) |u|^q dz\right) \\ \le \ \&\varepsilon c_{12} [u]^q + c_{11} \Vert u\Vert _r^r \text { for some } c_{12} > 0,\\ \Rightarrow \ \&\rho _a(Du) + \left( \frac{1}{2^{q - 1}} - \varepsilon c_{12}\right) [u]^q \le c_{11} \Vert u\Vert _r^r. \end{aligned}$$

Choosing $\varepsilon \in \left( 0,\displaystyle \frac{1}{2^{q - 1}c_{12}} \right)$, we obtain

$$\begin{aligned}&\rho _\theta (Du) + \int _\Omega \xi (z) |u|^q dz \le c_{13} \Vert u\Vert _r^r \text { for some } c_{13}> 0,\\ \Rightarrow \ \&\rho _\theta (Du) + \left( \int _\Omega \xi (z) |u|^q dz\right) ^{p/q} \le c_{14} \Vert u\Vert _r^r\\&\text { for some } c_{14}> 0\ (\text {recall that } |u| \le 1 \text { and that } q < p),\\ \Rightarrow \ \&\Vert Du\Vert _\theta ^p + \left( \int _\Omega \xi (z)\ |u|^q dz \right) ^{p/q} \le c_{14} \Vert u\Vert _r^r \ (\text {see Proposition 2}),\\ \Rightarrow \ \&|u|^p \le c_{15} \Vert u\Vert _r^r \text { for some } c_{15}> 0,\\ \Rightarrow \ \&\Vert u\Vert ^p \le c_{16} \Vert u\Vert _r^r \text { for some } c_{16} > 0. \end{aligned}$$

Therefore we conclude that we can find $\mu _0 > 0$ such that

$$\begin{aligned} 0 < \mu _0 \le \Vert u\Vert _r,\ \Vert u\Vert \text { for all } u \in \mathrm{N}. \end{aligned}$$

This proof is now complete. $\square$

From (8) it follows the following auxiliary property.

Corollary 13

If hypotheses $\mathrm{H_0},$ $\mathrm{H_1'}$ hold,  then

$$\begin{aligned} 0 < \mu _0 \le \Vert u\Vert _r,\ \Vert u\Vert \text { for all } u \in \mathrm{N_+}. \end{aligned}$$

Now let

$$\begin{aligned} m_+ = \inf \limits _\mathrm{N_+} \varphi _+ \text { and } m_- = \inf \limits _\mathrm{N_-} \varphi _-. \end{aligned}$$

We show that both infima are attained.

Proposition 14

If hypotheses $\mathrm{H_0},$ $\mathrm{H_1'}$ hold,  then there exist $u_0 \in \mathrm{N_+}$ and $v_0 \in \mathrm{N_-}$ such that

$$\begin{aligned} \varphi _+(u_0) = m_+ \text { and } \varphi _-(v_0) = m_-. \end{aligned}$$

Proof

We consider a sequence $\left\{ u_n\right\} _{n \in \mathbb {N}} \subseteq W^{1,\theta }(\Omega )$ such that

$$\begin{aligned} u_n \in \mathrm{N_+} \text { for all } n \in \mathbb {N}\text { and } \varphi _+(u_n) \downarrow m_+. \end{aligned}$$

(24)

From Corollary 7, we know that $\left\{ u_n\right\} _{n \in \mathbb {N}} \subseteq W^{1,\theta }(\Omega )$ is bounded and so by passing to a subsequence, we may assume that

$$\begin{aligned} u_n \xrightarrow w u_0 \text { in } W^{1,\theta }(\Omega ) \text { and } u_n \rightarrow u_0 \text { in } L^r(\Omega ). \end{aligned}$$

(25)

Since $\mathrm{N_+} \subseteq \mathrm{N}$ (see (8)), we have

$$\begin{aligned} \rho _a(Du_n) + \Vert Du_n\Vert _q^q + \int _\Omega \xi (z) |u_n|^q dz = \int _\Omega f(z,u_n) u_n dz,\ u_n \ge 0 \text { for all } n \in \mathbb {N}. \end{aligned}$$

(26)

Then $u_0 \ge 0$ (see (25)) and from Corollary 13 and (25), we see that $0 < \mu _0 \le \Vert u_0\Vert _r$, hence $u_0 \ne 0$.

From (25) and (26), we have

$$\begin{aligned} \rho _a(Du_0) + \Vert Du_0\Vert _q^q + \int _\Omega \xi (z) |u_0|^q dz \le \int _\Omega f(z,u_0) u_0 dz. \end{aligned}$$

Suppose that the above inequality is strict, that is,

$$\begin{aligned} \rho _a(Du_0) + \Vert Du_0\Vert _q^q + \int _\Omega \xi (z) |u_0|^q dz < \int _\Omega f(z,u_0) u_0 dz. \end{aligned}$$

(27)

We use the function $\beta _{u_0}(\cdot )$ from the proof of Proposition 8. We know that

$$\begin{aligned} \beta _{u_0}(t) > 0 \text { for } t \in (0,1) \text { small } \ (\text {see (18)}),\ \beta _{u_0}(1) < 0 \ (\text {see (15), (27)}). \end{aligned}$$

So, we can find $t_0 \in (0,1)$ such that

$$\begin{aligned}&\beta _{u_0}(t_0) = 0,\\ \Rightarrow \ \&t_0 k_{u_0}'(t_0) = 0 \text { since } u_0 \ge 0 \text { and so } \varphi _+(tu_0) = \varphi (tu_0),\ t \ge 0,\\ \Rightarrow \ \&t_0 u_0 \in \mathrm{N_+}. \end{aligned}$$

We have

$$\begin{aligned} m_+&\le \varphi _+(t_0u_0)\nonumber \\&= \frac{1}{p} \rho _a(D(t_0u_o)) + \frac{1}{q} \left( \Vert D(t_0u_0)\Vert _q^q + \int _\Omega \xi (z) |t_0u_0|^q dz\right) - \int _\Omega F(z,t_0u_0) dz\nonumber \\&= \frac{1}{p} \int _\Omega \left( f(z,t_0u_0)(t_0u_0) - pF(z,t_0u_0)\right) dz\nonumber \\&\quad + \left( \frac{1}{q} - \frac{1}{p}\right) \left( \Vert D(t_0u_0\Vert _q^q + \int _\Omega \xi (z)|t_0u_0|^q dz)\right) \ (\text {since } t_0u_0 \in \mathrm{N_+}). \end{aligned}$$

(28)

On account of (6), we have

$$\begin{aligned} f(z,sx) (sx) - pF(z,sx) \le f(z,x)x - pF(z,x) \end{aligned}$$

(29)

for a.a. $z \in \Omega$, all $x \in \mathbb {R}$, all $s \in (0,1)$.

Recall that $0< t_0 < 1$. So, using (27) and (29) in (28), we obtain

$$\begin{aligned} m_+&< \frac{1}{p} \int _\Omega (f(z,u_0)u_0 - pF(z,u_0)) dz\\&\quad + \left( \frac{1}{q} - \frac{1}{p}\right) \left( \Vert Du_0\Vert _q^q + \int _\Omega \xi (z) |u_0|^q dz\right) \\&\le \liminf \limits _{n \rightarrow \infty } \bigg (\frac{1}{p} \int _\Omega \left( f(z,u_n) u_n - pF(z,u_n)\right) dz\\&\quad + \left( \frac{1}{q} - \frac{1}{p}\right) \left( \Vert Du_n\Vert _q^q + \int _\Omega \xi (z) |u_n|^q dz\right) \bigg )\ (\text {see (25)})\\&= \liminf \limits _{n \rightarrow \infty } \left( \frac{1}{p}\rho _a(Du_n) + \frac{1}{q} \left( \Vert Du_n\Vert _q^q + \int _\Omega \xi (z) |u_n|^q dz \right) - \int _\Omega F(z,u_n) dz\right) \\&= \liminf \limits _{n \rightarrow \infty } \varphi _+(u_n) = m_+, \end{aligned}$$

a contradiction.

So, we have

$$\begin{aligned}&\rho _a(Du_0) + \Vert Du_0\Vert _q^q + \int _\Omega \xi (z) |u_0|^q dz = \int _\Omega f(z,u_0) u_0 dz,\\ \Rightarrow \ \&u_0 \in \mathrm{N_+}. \end{aligned}$$

Then it follows that

$$\begin{aligned} \rho _\theta (Du_n) \rightarrow \rho _\theta (Du_0) \ (\text {see (25)}). \end{aligned}$$

The uniform convexity of $\theta (z,\cdot )$ implies that

$$ \begin{aligned}&\rho _\theta (Du_n-Du_0) \rightarrow 0\ (\text {see Harjuletho } { \& } \,\,\mathrm{H}\ddot{\mathrm{a}}\mathrm{st}\ddot{\mathrm{o}} \text { [9, p. 65] }),\\ \Rightarrow \ \&Du_n \rightarrow Du_0 \text { in } L^\theta \left( \Omega ,\mathbb {R}^N \right) \ (\text {see Proposition 2}),\\ \Rightarrow \ \&u_n \rightarrow u_0 \text { in } W^{1,\theta }(\Omega )\ (\text {see(25) and Proposition 4}). \end{aligned}$$

So, finally we can say that

$$\begin{aligned} \varphi _+(u_0) = m_+,\ u_0 \in \mathrm{N_+}. \end{aligned}$$

Similarly, working with $\varphi _-(\cdot )$ and $\mathrm{N_-}$, we show that there exists $v_0 \in W^{1,\theta }(\Omega )$ such that

$$\begin{aligned} \varphi _-(v_0) = m_-,\ v_0 \in \mathrm{N_-}. \end{aligned}$$

This proof is now complete. $\square$

Next, we show that the two minimizers produced in Proposition 14 are in fact critical points of the functionals $\varphi _+$ and $\varphi _-$ respectively. This means that $\mathrm{N_+}$ and $\mathrm{N_-}$ are natural constraints for $\varphi _+$ and $\varphi _-$, respectively (see Papapgeorgiou et al. [19, p. 422]). Our proof is inspired by the works of Willem [26, p. 74] and Szulkin and Weth [25, p. 611].

Proposition 15

If hypothesis $\mathrm{H_0},$ $\mathrm{H_1'}$ hold and $u_0 \in \mathrm{N_+}$ and $v_0 \in \mathrm{N_-}$ are the two minimizers from Proposition 14,  then $u_0 \in K_{\varphi _+}$ and $v_0 \in K_{\varphi _-}.$

Proof

Let $\tau \in \mathring{\mathbb {R}}_+\!\setminus \!\{1\}$ ($\mathring{\mathbb {R}}_+ = (0,+\infty )$). Since $u_0 \in \mathrm{N_+}$ from Proposition 11 we have

$$\begin{aligned} \varphi _+(\tau u_0) \le \varphi _+(u_0) = m_+. \end{aligned}$$

(30)

We argue indirectly. So, suppose that $u_0 \not \in K_{\varphi _+}$. Then

$$\begin{aligned} \varphi _+'(u_0) \ne 0. \end{aligned}$$

We can find $\delta > 0$ and $d > 0$ such that

$$\begin{aligned} \Vert u - u_0\Vert \le 3\delta \Rightarrow \Vert \varphi _+'(u)\Vert _* \ge d. \end{aligned}$$

We consider the interval $I = \left( \displaystyle \frac{1}{2},\displaystyle \frac{3}{2}\right)$. From (30) we have

$$\begin{aligned} \hat{m}_+ = \max \limits _{\tau \in \partial I} \varphi _+(\tau u_0) < m_+. \end{aligned}$$

(31)

We can always choose $\delta > 0$ even smaller if necessary in order to have $\displaystyle \frac{d \delta }{8} \le \displaystyle \frac{m_+ - \hat{m}_+}{4}$. Then we apply the quantitative deformation theorem of Willem [26, p. 38] with $\varepsilon = \displaystyle \frac{d\delta }{8}$ and $S = \overline{B}_\delta (u_0) = \left\{ u \in W^{1,\theta }(\Omega ) : \Vert u - u_0\Vert \le \delta \right\}$. So, we produce a deformation $h : [0,1] \times W^{1,\theta }(\Omega ) \mapsto W^{1,\theta }(\Omega )$ such that

• $h(1,u) = u$ if $\varphi _+(u) \not \in [m_+ - 2\varepsilon , m_+ + 2\varepsilon ] \cap \overline{B}_{2\delta }(u_0)$;

• $h\left( 1,\varphi _+^{m_+ + \varepsilon } \cap \overline{B}_\delta (u_0)\right) \subseteq \varphi _+^{m_+ - \varepsilon }$;

• $\varphi _+(h(1,u)) \le \varphi _+(u) \text { for all } u \in W^{1,\theta }(\Omega )$.

From these properties of the deformation and (31), we deduce that

$$\begin{aligned} \max \limits _{\tau \in \overline{I}} \varphi _+(h(1,\tau u_0)) < m_+. \end{aligned}$$

(32)

Claim: $h(1,Iu_0) \cap \mathrm{N_+} \ne \emptyset$.

Let $\psi (\tau ) = h(1,\tau u_0)$ and consider the following two functions

$$\begin{aligned}&\sigma _0(\tau ) = \langle \varphi _+'(\tau u_0), u_0 \rangle ,\\&\sigma _1(\tau ) = \frac{1}{\tau } \langle \varphi _+'(\psi (\tau )),\psi (\tau ) \rangle ,\ \tau > 0. \end{aligned}$$

Recall that $u_0 \in \mathrm{N_+}$. So, we have

$$\begin{aligned}&\langle \varphi _+'(\tau u_0),u_0\rangle > 0 \text { if } 0< \tau < 1, \end{aligned}$$

(33)

$$\begin{aligned}&\langle \varphi _+'(\tau u_0),u_0\rangle< 0 \text { if } 1 < \tau \nonumber \\&(\text {see Propositions 11 and 9}). \end{aligned}$$

(34)

By $d_B(\cdot ,\cdot ,\cdot )$ we denote the Brouwer degree. Then from (33), (34) and Lloyd [13, see Illustration 1, pp. 19, 20], we have

$$\begin{aligned} d_B(\sigma _0,I,0) = -1. \end{aligned}$$

(35)

From the properties of the deformation, we have

$$\begin{aligned}&\sigma _0 |_{\partial I} = \sigma _1 |_{\partial I},\\ \Rightarrow \ \&d_B(\sigma _0,I,0) = d_B(\sigma _1,I,0),\\ \Rightarrow \ \&d_B(\sigma _1,I,0) \ne 0 \ (\text {see (35)}),\\ \Rightarrow \ \&\sigma _1(\tau ) = 0 \text { for some } \tau \in I \ (\text {see [9], [19]}),\\ \Rightarrow \ \&\psi (\tau ) = h(1,\tau u_0) \in \mathrm{N_+}. \end{aligned}$$

This proves the Claim.

But then on account of the Claim, we have

$$\begin{aligned} m_+ \le \max _{\tau \in \overline{I}} \varphi _+(h(1,\tau u_0)). \end{aligned}$$

Comparing with (32) we have a contradiction. Therefore $u_0 \in K_{\varphi _+}$.

Similarly we show that $v_0 \in K_{\varphi _-}$. $\square$

Now we are ready to produce two constant sign solutions for problem (1) (positive and negative solutions), under the strict monotonicity condition $\mathrm{H_1'(iii)}$.

Proposition 16

If hypotheses $\mathrm{H_0},$ $\mathrm{H_1'}$ hold,  then problem (1) has at least two nontrivial constant sign solutions

$$\begin{aligned}&u_0 \in \mathrm{N_+} \cap L^\infty (\Omega ),\ u_0(z) > 0 \text { for a.a. } z \in \Omega ,\\&v_0 \in \mathrm{N_-}\cap L^\infty (\Omega ),\ v_0(z) < 0 \text { for a.a. } z \in \Omega . \end{aligned}$$

Proof

From Proposition 15 we know that

$$\begin{aligned} u_0 \in \mathrm{N_+} \text { and } v_0 \in \mathrm{N_-} \end{aligned}$$

are nontrivial constant sign solutions of problem (1). As in Gasiński and Winkert [6, Theorem 3.1], we have that

$$\begin{aligned} u_0 \in \mathrm{N_+} \cap L^\infty (\Omega ),\ v_0 \in \mathrm{N_-} \cap L^\infty (\Omega ). \end{aligned}$$

Finally using Proposition 2.4 of Papageorgiou, Vetro and Vetro [22], we have

$$\begin{aligned} v_0(z)< 0 < u_0(z) \text { for a.a. } z \in \Omega . \end{aligned}$$

This proof is now complete. $\square$

4 Nodal solutions under strict monotonicity

In this section we prove the existence of a nodal (that is, sign changing) solution for problem (1), under the strict monotonicity hypothesis $\mathrm{H_1'(iii)}$. For this purpose, we bring in the picture the submanifold $\mathrm{N_0}$.

We define

$$\begin{aligned} m_0 = \inf \limits _\mathrm{N_0} \varphi . \end{aligned}$$

Proposition 17

If hypotheses $\mathrm{H_0},$ $\mathrm{H_1'}$ hold,  then there exists $y_0 \in \mathrm{N_0}$ such that

$$\begin{aligned} m_0 = \varphi (y_0). \end{aligned}$$

Proof

We consider a sequence $\left\{ y_n\right\} _{n \in \mathbb {N}} \subseteq W^{1,\theta }(\Omega )$ such that

$$\begin{aligned} y_n \in \mathrm{N_0} \text { for all } n \in \mathbb {N}\text { and } \varphi (y_n) \downarrow m_0. \end{aligned}$$

From Proposition 6 we know that $\left\{ y_n\right\} _{n \in \mathbb {N}} \subseteq W^{1,\theta }(\Omega )$ is bounded. Then $\left\{ y_n^+\right\} _{n \in \mathbb {N}} \subseteq W^{1,\theta }(\Omega )$ and $\left\{ y_n^-\right\} _{n \in \mathbb {N}} \subseteq W^{1,\theta }(\Omega )$ are bounded, too. So, we may assume that

$$\begin{aligned} y_n^+ \xrightarrow w \hat{y}_1 \text { and } y_n^- \xrightarrow w \hat{y}_2 \text { in } W^{1,\theta }(\Omega ). \end{aligned}$$

(36)

Suppose that $\hat{y}_1 = 0$. Since $y_n \in \mathrm{N}$, we have that $y_n^+ \in \mathrm{N}_+$ for all $n \in \mathbb {N}$ and so

$$\begin{aligned} \rho _a(Dy_n^+) + \Vert Dy_n^+\Vert _q^q + \int _\Omega \xi (z) (y_n^+)^q dz&= \int _\Omega f(z,y_n^+)y_n^+ dz \text { for all } n \in \mathbb {N}, \nonumber \\ \Rightarrow \ \&\rho _\theta (Dy_n^+) + \int _\Omega \xi (z) (y_n^+)^q dz \rightarrow 0\nonumber \\&\ (\text {see (36) and recall that } \hat{y}_1 = 0),\nonumber \\ \Rightarrow \ \&|y_n^+| \rightarrow 0, \nonumber \\ \Rightarrow \ \&y_n^+ \rightarrow 0 \text { in } W^{1,\theta }(\Omega ) \ (\text {see Proposition 4}). \end{aligned}$$

(37)

But from Corollary 13 we know that

$$\begin{aligned} 0 < \mu _0 \le \Vert y_n^+\Vert \text { for all } n \in \mathbb {N}. \end{aligned}$$

(38)

Comparing (37) and (38), we have a contradiction.

Therefore $\hat{y}_1 \ne 0$. Similarly we show that $\hat{y}_2 \ne 0$.

According to Proposition 9, there exist a unique $\hat{t}_1$, $\hat{t}_2 > 0$ such that

$$\begin{aligned} \hat{t}_1 \hat{y}_1 \in \mathrm{N_+} \text { and }\hat{t}_2 \hat{y}_2 \in \mathrm{N_-}. \end{aligned}$$

We set

$$\begin{aligned} y_0 = \hat{t}_1 \hat{y}_1 - \hat{t}_2 \hat{y}_2. \end{aligned}$$

Evidently we have

$$\begin{aligned} y_0^+ = \hat{t}_1 \hat{y}_1,\ y_0^- = \hat{t}_2 \hat{y}_2 \text { and so } y_0 \in \mathrm{N_0}. \end{aligned}$$

We have

$$\begin{aligned} m_0&= \lim \limits _{n \rightarrow \infty } \varphi (y_n)\\&= \lim \limits _{n\rightarrow \infty } \left( \varphi _+(y_n^+) + \varphi _-(-y_n^-)\right) \\&\ge \liminf \limits _{n \rightarrow \infty } \left( \varphi _+(\hat{t}_1y_n^+) + \varphi _-(-\hat{t}_2y_n^-)\right) \ (\text {see Proposition 11})\\&\ge \varphi (y_0)\\&\ge m_0\ (\text {since } y_0 \in \mathrm{N_0}),\\ \Rightarrow \ \&m_0 = \varphi (y_0) \text { with } y_0 \in \mathrm{N_0}. \end{aligned}$$

This proof is now complete. $\square$

Reasoning as in the proof of Proposition 15, we show that $\mathrm{N_0}$ is a natural constraint for $\varphi (\cdot )$, that is, $y_0 \in K_\varphi$. In this case we apply the quantitative deformation theorem of Willem [26], using the following data: the set

$$\begin{aligned} D = \left( \frac{1}{2},\frac{3}{2}\right) \times \left( \frac{1}{2},\frac{3}{2}\right) \end{aligned}$$

and the functions

$$\begin{aligned}&\psi (\tau ,t) = h(1,\tau \hat{y}_1 - t \hat{y}_2),\ \tau ,\ t \ge 0,\\&\sigma _0(\tau ,t) = \left( \langle \varphi '(\tau \hat{y}_1),\hat{y}_1 \rangle ,\langle \varphi '(-t\hat{y}_2),-\hat{y}_2\rangle \right) \\&\sigma _1(\tau ,t) = \left( \frac{1}{\tau } \langle \varphi '(\psi (\tau ,t)^+),\psi (\tau ,t)^+\rangle ,\frac{1}{t} \langle \varphi '(-\psi (\tau ,t)^-),-\psi (\tau ,t)^-\rangle \right) . \end{aligned}$$

Using the quantitative deformation theorem of Willem [26, p. 38], with the above data, we have the following result.

Proposition 18

If hypotheses $\mathrm{H_0},$ $\mathrm{H_1'}$ hold and $y_0 \in \mathrm{N_0}$ is the minimizer from Proposition 17,  then $y_0 \in K_\varphi .$

So, under the stronger strict monotonicity condition, we can state the following multiplicity theorem for problem (1).

Proposition 19

If hypotheses $\mathrm{H_0},$ $\mathrm{H_1'}$ hold,  then problem (1) has at least three nontrivial solutions

$$\begin{aligned}&u_0 \in \mathrm{N_+} \cap L^\infty (\Omega ),\ u_0(z) > 0 \text { for a.a. } z \in \Omega ,\ m_+ = \varphi _+(u_0),\\&v_0 \in \mathrm{N_-} \cap L^\infty (\Omega ),\ v_0(z) < 0 \text { for a.a. } z \in \Omega ,\ m_- = \varphi _-(v_0),\\&y_0 \in \mathrm{N_0} \cap L^\infty (\Omega ) \text { nodal},\ m_0 = \varphi (y_0). \end{aligned}$$

5 Multiple solutions under relaxed monotonicity

In this section we drop the strict monotonicity condition $\mathrm{H_1'(iii)}$ and prove a multiplicity result (three solutions theorem) with sign information for all the solutions, under the weaker set of hypotheses $\mathrm{H_1}$.

For this purpose, we introduce the following perturbation of the reaction

$$\begin{aligned} f_\ell (z,x) = f(z,x) + \ell |x|^{r - 2} x,\ \ell > 0. \end{aligned}$$

This perturbation of $f(\cdot ,\cdot )$ satisfies hypotheses $\mathrm{H_1'}$. We set $F_\ell (z,x) = \displaystyle \int _0^x f_\ell (z,s) ds$ and we consider the corresponding energy functionals $\varphi _\ell$, $\varphi _\ell ^\pm : W^{1,\theta }(\Omega ) \mapsto \mathbb {R}$ defined as $\varphi$, $\varphi _\pm$ only now F(z, x) is replaced by $F_\ell (z,x)$. Also, to these functionals we associate the corresponding Nehari manifolds $\mathrm{N}^\ell$, $\mathrm{N}_\pm ^\ell , \mathrm{N}_0^\ell$.

Proposition 20

If hypotheses $\mathrm{H_0},$ $\mathrm{H_1}$ hold,  then there exists $\hat{\mu } > 0$ such that

$$\begin{aligned} 0 < \hat{\mu } \le \varphi _\ell (u) \text { for all } u \in \mathrm{N}^\ell , \text { all } \ell \in (0,1). \end{aligned}$$

Proof

On account of hypotheses $\mathrm{H_1(i)}$, (iv), given $\varepsilon > 0$, we can find $c_{17} = c_{17}(\varepsilon ) > 0$ such that

$$\begin{aligned} F(z,x) \le \frac{\varepsilon }{q} |x|^q + c_{17} |x|^r \text { for a.a. } z \in \Omega , \text { all } x \in \mathbb {R}. \end{aligned}$$

(39)

Let $u \in \mathrm{N}^\ell$ ($\ell \in (0,1)$). From Proposition 10, we have

$$\begin{aligned} \varphi _\ell (u)&= \max \limits _{t \ge 0} \varphi _\ell (tu)\\&\ge \max \limits _{t \ge 0} \bigg (\frac{t^p}{p} \rho _a(Du) + \frac{t^q}{q}\left( \Vert Du\Vert _q^q + \int _\Omega \xi (z) |u|^q dz\right) \\&- \int _\Omega F(z,tu) dz - t^r\Vert u\Vert _r^r\bigg )\ (\text {since } 0< \ell < 1). \end{aligned}$$

Using (39), Proposition 5 and choosing $\varepsilon > 0$ small, we obtain

$$\begin{aligned}&\varphi _\ell (u) \ge \max \limits _{t \ge 0} \left( c_{18} t^p - c_{19} t^r\right) \text { for some } c_{18},\ c_{19}> 0, \text { all } t \ge 0,\\ \Rightarrow \ \&\varphi _\ell (u) \ge \hat{\mu } > 0\ (\text {recall that } p < r). \end{aligned}$$

This proof is now complete. $\square$

In a similar fashion we show similar results for the functionals $\varphi _\ell ^\pm (\cdot )$.

Proposition 21

If hypotheses $\mathrm{H_0},$ $\mathrm{H_1}$ hold,  then there exist $\hat{\mu }_\pm > 0$ such that

$$\begin{aligned}&0< \hat{\mu }_+ \le \varphi _\ell ^+(u) \text { for all } u \in \mathrm{N}_+^\ell , \text { all } \ell \in (0,1),\\&0 < \hat{\mu }_- \le \varphi _\ell ^-(u) \text { for all } u \in \mathrm{N}_-^\ell , \text { all } \ell \in (0,1). \end{aligned}$$

Now letting $\ell \rightarrow 0^+$ we will produce constant sign and nodal solutions for problem (1) under the relaxed monotonicity condition $\mathrm{H_1(iii)}.$

Theorem 22

If hypotheses $\mathrm{H_0},$ $\mathrm{H_1}$ hold,  then problem (1) has at least three nontrivial solutions

$$\begin{aligned}&u_0 \in \mathrm{N_+} \cap L^\infty (\Omega ),\ u_0(z) > 0 \text { for a.a. } z \in \Omega ,\ \varphi _+(u_0) = m_+,\\&v_0 \in \mathrm{N_-} \cap L^\infty (\Omega ),\ v_0(z) < 0 \text { for a.a. } z \in \Omega ,\ \varphi _-(v_0) = m_- ,\\&y_0 \in \mathrm{N_0} \cap L^\infty (\Omega ) \text { nodal},\ \varphi (y_0) =\ m_0. \end{aligned}$$

Proof

Let $\ell _n \rightarrow 0^+$. From Proposition 19 we know that we can find $y_n \in W^{1,\theta }(\Omega )$ such that

$$\begin{aligned}&y_n \in \mathrm{N}^{\ell _n}\ (n \in \mathbb {N}), \ \varphi _{\ell _n}(y_n) = m_0^{\ell _n} \rightarrow \hat{m}_0 \ge 0 \text { as } n \rightarrow \infty ,\ \varphi _{\ell _n}'(y_n) = 0\ (n \in \mathbb {N}) \nonumber \\&(\text {note that } \left\{ m_0^{\ell _n}\right\} _{n \in \mathbb {N}} \subseteq \mathring{\mathbb {R}}_+ \text { is decreasing}). \end{aligned}$$

(40)

We will show that $\left\{ y_n\right\} _{n \in \mathbb {N}} \subseteq W^{1,\theta }(\Omega )$ is bounded. Arguing by contradiction, suppose that at least for a subsequence, we have

$$\begin{aligned} \Vert y_n\Vert \rightarrow + \infty . \end{aligned}$$

(41)

We set $\hat{w}_n = \displaystyle \frac{y_n}{\Vert y_n\Vert }$ ($n \in \mathbb {N}$). Then $\Vert \hat{w}_n\Vert = 1$ for all $n \in \mathbb {N}$ and so we may assume that

$$\begin{aligned} \hat{w}_n \xrightarrow w \hat{w} \text { in } W^{1,\theta }(\Omega ) \text { and } \hat{w}_n \rightarrow \hat{w} \text { in } L^r(\Omega ). \end{aligned}$$

(42)

Suppose that $\hat{w} \equiv 0$ and let $k > 1$. Then on account of Proposition 11, we have

$$\begin{aligned} m_0^{\ell _n}&= \varphi _{\ell _n}(y_n)\\&\ge \varphi _{\ell _n}\left( \frac{k}{\Vert y_n\Vert } y_n\right) \\&= \varphi _{\ell _n}(k \hat{w}_n)\\&= \frac{k^p}{p} \rho _a(D\hat{w}_n) + \frac{k^q}{q} \left( \Vert D\hat{w}_n\Vert _q^q + \int _\Omega \xi (z) |\hat{w}_n|^q dz\right) \\&\quad - \int _\Omega F(z,k\hat{w}_n) dz - \ell k^r\Vert \hat{w}_n\Vert _r^r \\ \ \ \ \ \ \&\ge \frac{k^q}{p} c_{20} - \int _\Omega F(z,k\hat{w}_n) dz - \ell k^r \Vert \hat{w}_n\Vert _r^r\\&\text { for some } c_{20}> 0\ (\text {since } k > 1,\ q < p,\ \Vert \hat{w}_n\Vert = 1). \end{aligned}$$

Passing to the limit as $n \rightarrow \infty$ and recalling that $\hat{w} = 0$, we obtain

$$\begin{aligned} \hat{m}_0 \ge \frac{k^q}{p} c_{20}\ (\text {see (40), (42)}). \end{aligned}$$

Since $k > 1$ is arbitrary, we let $k \rightarrow + \infty$ and have a contradiction.

Now assume that $\hat{w} \ne 0$ and let $\hat{\Omega } = \left\{ z \in \Omega : \hat{w}(z) \ne 0\right\}$.

Then $|\hat{\Omega }|_N > 0$ (by $|\cdot |_N$ we denote the Lebesgue measure on $\mathbb {R}^N$). We have

$$\begin{aligned} |y_n(z)| \rightarrow + \infty \text { for a.a. } z \in \hat{\Omega }. \end{aligned}$$

From (40) we have

$$\begin{aligned} 0&\le \frac{m_0^{\ell _n}}{\Vert y_n\Vert ^p}\nonumber \\&=\frac{\varphi _{\ell _n}(y_n)}{\Vert y_n\Vert ^p}\nonumber \\&\le \frac{1}{p} \rho _a(D\hat{w}_n) + \frac{1}{q \Vert y_n\Vert ^{p - q}} \left( \Vert D\hat{w}_n\Vert _q^q + \int _\Omega \xi (z) |\hat{w}_n|^q dz\right) - \int _\Omega \frac{F(z,y_n)}{\Vert y_n\Vert ^p} dz\nonumber \\&\le \frac{1}{q} \left( \rho _\theta (D\hat{w}_n) + \int _\Omega \xi (z) |\hat{w}_n|^q dz\right) - \int _\Omega \frac{F(z,y_n)}{\Vert y_n\Vert ^p} dz \text { for all } n \ge n_1\nonumber \\&\le \frac{1}{q} \left( \Vert D\hat{w}_n\Vert _\theta ^q + \int _\Omega \xi (z) |\hat{w}_n|^q dz\right) - \int _\Omega \frac{F(z,y_n)}{\Vert y_n\Vert ^p} dz\nonumber \\&\ \ \ (\text {since } \Vert D\hat{w}_n\Vert _\theta \le \Vert \hat{w}_n\Vert = 1, \text { see Proposition 2})\nonumber \\&\le \frac{1}{q} c_{21} - \int _\Omega \frac{F(z,y_n)}{\Vert y_n\Vert ^p} dz \nonumber \\&\ \ \text { for some } c_{21} > 0, \text { all } n \ge n_1\ (\text {see Proposition 4}). \end{aligned}$$

(43)

On account of hypothesis $\mathrm{H_1(ii)}$, we have

$$\begin{aligned}&\int _{\hat{\Omega }} \frac{F(z,y_n)}{\Vert y_n\Vert ^p} dz \rightarrow + \infty ,\nonumber \\ \Rightarrow \ \&\int _\Omega \frac{F(z,y_n)}{\Vert y_n\Vert ^p} dz \rightarrow + \infty \nonumber \\&(\text {see } \mathrm{H_1(ii)} \text { and use Fatou's lemma}). \end{aligned}$$

(44)

If in (43) we pass to the limit as $n \rightarrow \infty$ and use (44) we have a contradiction.

Therefore $\left\{ y_n\right\} _{n \in \mathbb {N}} \subseteq W^{1,\theta }(\Omega )$ is bounded and so we may assume that

$$\begin{aligned} y_n \xrightarrow w y_0 \text { in } W^{1,\theta }(\Omega ) \text { and } y_n \rightarrow y_0 \text { in } L^r(\Omega ). \end{aligned}$$

(45)

From (40) we have

$$\begin{aligned} \langle V(y_n), h\rangle + \int _\Omega \xi (z) |y_n|^{q - 2} y_n h dz = \int _\Omega f(z,y_n)h dz \text { for all } h \in W^{1,\theta }(\Omega ). \end{aligned}$$

We choose $h = y_n - y_0 \in W^{1,\theta }(\Omega )$, pass to the limits as $n \rightarrow \infty$ and use (45). We obtain

$$\begin{aligned}&\lim \limits _{n \rightarrow \infty } \langle V(y_n), y_n - y_0\rangle =0,\nonumber \\ \Rightarrow \ \&y_n \rightarrow y_0 \text { in } W^{1,\theta }(\Omega )\ (\text {see Proposition 3}). \end{aligned}$$

(46)

From (46) and Proposition 20, we have

$$\begin{aligned}&0 < \hat{\mu } \le \varphi (y_0),\\ \Rightarrow \ \&y_0 \ne 0 \text { and so } y_0 \in \mathrm{N}. \end{aligned}$$

Reasoning as in the proof Proposition 17, using Proposition 21, we show that $y_0^+ \ne 0$, $y_0^- \ne 0$, hence $y_0 \in \mathrm{N}_0$. Therefore we have

$$\begin{aligned}&y_0 \in \mathrm{N_0}, \ \varphi '(y_0) = 0,\ \varphi (y_0) = m_0,\\ \Rightarrow \ \&y_0 \in \mathrm{N} \cap L^\infty (\Omega ) \text { is a ground state nodal solution of problem (1)}. \end{aligned}$$

Similarly, working with $\varphi _{\ell _n}^+$ and $\mathrm{N}_+^{\ell _n}$ and passing to the limit as $n \rightarrow \infty$, we obtain $u_0 \in W^{1,\theta }(\Omega )$ such that

$$\begin{aligned} u_0 \in \mathrm{N_+} \cap L^\infty (\Omega ),\ \varphi '(u_0) = 0,\ u_0(z) > 0 \text { for a.a. } z \in \Omega ,\ \varphi (u_0) = m_+ \end{aligned}$$

(positive ground state solution) and $v_0 \in W^{1,\theta }(\Omega )$ such that

$$\begin{aligned} v_0 \in \mathrm{N_-} \cap L^\infty (\Omega ),\ \varphi '(v_0) = 0,\ v_0(z) < 0 \text { for a.a. } z \in \Omega ,\ \varphi (v_0) = m_- \end{aligned}$$

(negative ground state solution). $\square$