Singular Dirichlet (p, q)-Equations

Abstract

We consider a nonlinear Dirichlet problem driven by the (p, q)-Laplacian and with a reaction having the combined effects of a singular term and of a parametric $(p-1)$-superlinear perturbation. We prove a bifurcation-type result describing the changes in the set of positive solutions as the parameter $\lambda >0$ varies. Moreover, we prove the existence of a minimal positive solution $u^*_\lambda$ and study the monotonicity and continuity properties of the map $\lambda \rightarrow u^*_\lambda$.

1 Introduction

In a recent paper, the authors [15] studied the following singular parametric p-Laplacian Dirichlet problem

$$\begin{aligned} -\Delta _p u= & {} u^{-\eta } +\lambda f(x,u)\quad \text {in } \Omega ,\nonumber \\ u= & {} 0 \quad \text {on } \partial \Omega ,\nonumber \\ u> & {} 0, \quad \lambda >0, \quad 0<\eta<1, \quad 1<p. \end{aligned}$$

They proved a result describing the dependence of the set of positive solutions as the parameter $\lambda >0$ varies, assuming that $f(x,\cdot )$ is $(p-1)$-superlinear.

In the present paper, we consider a singular parametric Dirichlet problem driven by the (p, q)-Laplacian, that is, the sum of a p-Laplacian and of a q-Laplacian with $1<q<p$. To be more precise, the problem under consideration is the following

where $\Omega \subseteq \mathbb {R}^N$ is a bounded domain with a $C^2$-boundary $\partial \Omega$. In this problem, the differential operator is not homogeneous and so many of the techniques used in Papageorgiou–Winkert [15] are not applicable here. More precisely, in the proof of Proposition 3.1 in [15], the homogeneity of the p-Laplacian is crucial in the argument. It provides naturally an upper solution $\overline{u}$ which is an appropriate multiple of the unique solution $e \in {{\,\mathrm{int}\,}}\left( C^1_0(\overline{\Omega })_+\right)$ of problem (3.2) in [15] (see also the argument in (3.7)). In our setting, this is no longer possible since the differential operator, the (p, q)-Laplacian, is not homogeneous. This makes our proof here of the fact that $\mathcal {L} \ne \emptyset$ (existence of admissible parameters, see Proposition 3.1) more involved and requires some preparation which involves Propositions 2.3 and 2.4. Moreover, the proof that the critical parameter $\lambda ^*>0$ is finite differs for the same reason and here is more involved and requires the use of a different strong comparison principle. In [15] (see Proposition 3.6) this is done easily since we can use the spectrum of $(-\Delta _p,W^{1,p}_0(\Omega ))$ and in particular the principal eigenvalue $\hat{\lambda }_1>0$ thanks to the homogeneity of the differential operator (see (3.25) in [15]). This reasoning fails in our setting and leads to a different geometry near zero (compare hypothesis H(iv) in [15] with hypothesis H(iv) in this paper). Furthermore, we now need to employ a different comparison argument based on a recent strong comparison principle due to Papageorgiou–Rădulescu–Repovš [12]. In addition, the proof of Proposition 3.7 in [15] cannot be extended to our problem (see the part from (3.42) and below). The presence of the q-Laplacian leads to difficulties. For this reason, our superlinearity condition (see hypothesis H(iii)) differs from the one used in [15]. However, we stress that both go beyond the classical Ambrosetti–Rabinowitz condition.

For the parametric perturbation of the singular term, $\lambda f(\cdot ,\cdot )$ with $f:\Omega \times \mathbb {R}\rightarrow \mathbb {R}$, we assume that f is a Carathéodory function, that is, $x\mapsto f(x,s)$ is measurable for all $s\in \mathbb {R}$ and $s\mapsto f(x,s)$ is continuous for almost all (a. a.) $x\in \Omega$. Moreover we assume that $f(x,\cdot )$ exhibits $(p-1)$-superlinear growth as $s\rightarrow +\infty$ but it need not satisfy the usual Ambrosetti–Rabinowitz condition (the AR-condition for short) in such cases. Applying variational tools from critical point theory along with suitable truncation and comparison techniques, we prove a bifurcation-type result as in [15], which describes in a precise way the dependence of the set of positive solutions as the parameter $\lambda >0$ changes.

In this direction we mention the recent works of Papageorgiou–Rădulescu–Repovš [12] and Papageorgiou–Vetro–Vetro [14] which also deal with nonlinear singular parametric Dirichlet problems. In theses works the parameter multiplies the singular term. Indeed, in Papageorgiou–Rădulescu–Repovš [12] the equation is driven by a nonhomogeneous differential operator and in the reaction we have the competing effects of a parametric singular term and of a $(p-1)$-superlinear perturbation. In Papageorgiou–Vetro–Vetro [14] the equation is driven by the (p, 2)-Laplacian and in the reaction we have the competing effects of a parametric singular term and of a $(p-1)$-linear, resonant perturbation. The work of Papageorgiou–Vetro–Vetro [14] was continued by Bai–Motreanu–Zeng [2] where the authors examine the continuity properties with respect to the parameter of the solution multifunction.

Boundary value problems monitored by a combination of differential operators of different nature (such as (p, q)-equations), arise in many mathematical processes. We refer, for example, to the works of Bahrouni–Rădulescu–Repovš [1] (transonic flows), Benci–D’Avenia–Fortunato–Pisani [3] (quantum physics), Cherfils–Il$'$yasov [4] (reaction diffusion systems) and Zhikov [19] (elasticity theory). We also mention the survey paper of Rădulescu [18] on anisotropic (p, q)-equations.

2 Preliminaries and Hypotheses

The main spaces which we will be using in the study of problem (P$_\lambda$) are the Sobolev space $W^{1,p}_0(\Omega )$ and the Banach space $C^1_0(\overline{\Omega })$. By $\Vert \cdot \Vert$ we denote the norm of the Sobolev space $W^{1,p}_0(\Omega )$ and because of the Poincaré inequality, we have

$$\begin{aligned} \Vert u\Vert =\Vert \nabla u\Vert _p \quad \text {for all }u\in W^{1,p}_0(\Omega ), \end{aligned}$$

where $\Vert \cdot \Vert _p$ denotes norm in $L^{p}(\Omega )$ and also in $L^p(\Omega ;\mathbb {R}^N)$. From the context it will be clear which one is used.

The Banach space

$$\begin{aligned} C^1_0(\overline{\Omega })= \left\{ u \in C^1(\overline{\Omega })\,:\, u\big |_{\partial \Omega }=0 \right\} \end{aligned}$$

is an ordered Banach space with positive cone

$$\begin{aligned} C^1_0(\overline{\Omega })_+=\left\{ u \in C^1_0(\overline{\Omega })\,:\, u(x) \ge 0 \text { for all } x \in \overline{\Omega }\right\} . \end{aligned}$$

This cone has a nonempty interior given by

$$\begin{aligned} {{\,\mathrm{int}\,}}\left( C^1_0(\overline{\Omega })_+\right) =\left\{ u \in C^1_0(\overline{\Omega })_+: u(x)>0 \text { for all } x \in \Omega \text {, } \frac{\partial u}{\partial n}(x)<0 \text { for all } x \in \partial \Omega \right\} , \end{aligned}$$

where $n(\cdot )$ stands for the outward unit normal on $\partial \Omega$.

For every $r\in (1,\infty )$, let $A_r:W^{1,r}_0(\Omega )\rightarrow W^{-1,r'}(\Omega )=W^{1,r}_0(\Omega )^*$ with $\frac{1}{r}+\frac{1}{r'}=1$ be the nonlinear map defined by

$$\begin{aligned} \langle A_r(u), h\rangle =\int _{\Omega }|\nabla u|^{r-2}\nabla u \cdot \nabla h\,\mathrm{d}x \quad \text {for all }u,h\in W^{1,r}_0(\Omega ). \end{aligned}$$

(2.1)

From Gasiński-Papageorgiou [5, Problem 2.192, p. 279] we have the following properties of $A_r$.

Proposition 2.1

The map $A_r:W^{1,r}_0(\Omega )\rightarrow W^{-1,r'}(\Omega )$ defined in (2.1) is bounded, that is, it maps bounded sets to bounded sets, continuous, strictly monotone, hence maximal monotone and it is of type $({{\,\mathrm{S}\,}})_+$, that is,

$$\begin{aligned} u_n \rightharpoonup u \text { in }W^{1,r}_0(\Omega )\quad \text {and}\quad \limsup _{n\rightarrow \infty } \langle A_r(u_n),u_n-u\rangle \le 0, \end{aligned}$$

imply $u_n\rightarrow u$ in $W^{1,r}_0(\Omega )$.

For $s \in \mathbb {R}$, we set $s^{\pm }=\max \{\pm s,0\}$ and for $u \in W^{1,p}_0(\Omega )$ we define $u^{\pm }(\cdot )=u(\cdot )^{\pm }$. It is well known that

$$\begin{aligned} u^{\pm } \in W^{1,p}_0(\Omega ), \quad |u|=u^++u^-, \quad u=u^+-u^-. \end{aligned}$$

For $u,v\in W^{1,p}_0(\Omega )$ with $u(x)\le v(x)$ for a. a. $x\in \Omega$ we define

$$\begin{aligned} {[}u,v]&=\big \{h\in W^{1,p}_0(\Omega ): u(x)\le h(x)\le v(x)\text { for a. a. }x\in \Omega \big \},\\ {[}u)&=\big \{h\in W^{1,p}_0(\Omega ): u(x)\le h(x)\text { for a. a. }x\in \Omega \big \}. \end{aligned}$$

Given a set $S\subseteq W^{1,p}(\Omega )$ we say that it is “downward directed”, if for any given $u_1, u_2\in S$ we can find $u \in S$ such that $u\le u_1$ and $u\le u_2$.

If $h_1,h_2:\Omega \rightarrow \mathbb {R}$ are two measurable functions, then we write $h_1\prec h_2$ if and only if for every compact $K\subseteq \Omega$ we have $0<c_K\le h_2(x)-h_1(x)$ for a. a. $x\in K$.

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

$$\begin{aligned} K_\varphi =\left\{ u\in X \, : \, \varphi '(u)=0\right\} \end{aligned}$$

being the critical set of $\varphi$. Furthermore, we say that $\varphi$ satisfies the Cerami condition (C-condition for short), if every sequence $\{u_n\}_{n \ge 1} \subseteq X$ such that $\{\varphi (u_n)\}_{n \ge 1}\subseteq \mathbb {R}$ is bounded and such that $\left( 1+\Vert u_n\Vert _X\right) \varphi '(u_n) \rightarrow 0$ in $X^*$ as $n \rightarrow \infty$, admits a strongly convergent subsequence.

Our Hypotheses on the perturbation $f:\Omega \times \mathbb {R}\rightarrow \mathbb {R}$ are the following:

• H: $f:\Omega \times \mathbb {R}\rightarrow \mathbb {R}$ is a Carathéodory function such that $f(x,0)=0$ for a. a. $x\in \Omega$ and

  • (i)
    $$\begin{aligned} f(x,s)\le a(x) \left( 1+s^{r-1}\right) \end{aligned}$$

    for a.a. $x\in \Omega$, for all $s\ge 0$, with $a\in L^{\infty }(\Omega )$ and $p<r<p^*$, where $p^*$ denotes the critical Sobolev exponent with respect to p given by

    $$\begin{aligned} p^*= {\left\{ \begin{array}{ll} \frac{Np}{N-p} &{} \text {if }p<N,\\ +\infty &{} \text {if } N \le p; \end{array}\right. } \end{aligned}$$

  • (ii) if $F(x,s)=\int ^s_0f(x,t)\mathrm{d}t$, then

    $$\begin{aligned} \lim _{s\rightarrow +\infty } \frac{F(x,s)}{s^{p}}=+\infty \quad \text {uniformly for a.\,a.\,}x\in \Omega ; \end{aligned}$$

  • (iii) there exists $\tau \in \left( (r-p)\max \left\{ \frac{N}{p},1\right\} ,p^*\right)$ with $\tau >q$ such that

    $$\begin{aligned} 0 < c_0\le \liminf _{s\rightarrow +\infty } \frac{f(x,s)s-pF(x,s)}{s^\tau } \quad \text {uniformly for a.\,a.\,}x\in \Omega ; \end{aligned}$$
  • (iv)
    $$\begin{aligned} \lim _{s\rightarrow 0^+} \frac{f(x,s)}{s^{q-1}}=0\quad \text {uniformly for a. a. }x\in \Omega \end{aligned}$$

    and there exists $\tau \in (q,p)$ such that

    $$\begin{aligned} \liminf _{s\rightarrow 0^+}\, \frac{f(x,s)}{s^{\tau -1}}\ge \hat{\eta }>0\quad \text {uniformly for a. a. }x\in \Omega ; \end{aligned}$$

  • (v) for every $\hat{s}>0$ we have

    $$\begin{aligned} f(x,s) \ge m_{\hat{s}}>0 \end{aligned}$$

    for a.a. $x\in \Omega$ and for all $s\ge \hat{s}$ and for every $\rho >0$ there exists $\hat{\xi }_\rho >0$ such that the function

    $$\begin{aligned} s\rightarrow f(x,s)+\hat{\xi }_\rho s^{p-1} \end{aligned}$$

    is nondecreasing on $[0,\rho ]$ for a.a. $x\in \Omega$.

Remark 2.2

Since we are looking for positive solutions and the hypotheses above concern the positive semiaxis $\mathbb {R}_+=[0,+\infty )$, without any loss generality, we may assume that

$$\begin{aligned} f(x,s)=0\quad \text {for a.a.\,}x\in \Omega \text { and for all }s\le 0. \end{aligned}$$

(2.2)

Hypotheses H(ii), H(iii) imply that

$$\begin{aligned} \lim _{s\rightarrow +\infty } \frac{f(x,s)}{s^{p-1}}=+\infty \quad \text {uniformly for a.a. }x\in \Omega . \end{aligned}$$

Hence, the perturbation $f(x,\cdot )$ is $(p-1)$-superlinear. In the literature, superlinear equations are usually treated using the AR-condition. In our case, taking (2.2) into account, we refer to a unilateral version of this condition which says that there exist $M>0$ and $\mu >p$ such that

$$\begin{aligned} 0&<\mu F(x,s) \le f(x,s)s\quad \text {for a. a. }x\in \Omega \text { and for all }s\ge M, \end{aligned}$$

(2.3)

$$\begin{aligned} 0&<{{\,\mathrm{ess~inf}\,}}_\Omega F(\cdot ,M). \end{aligned}$$

(2.4)

If we integrate (2.3) and use (2.4), we obtain the weaker condition

$$\begin{aligned} c_1 s^\mu \le F(x,s)\quad \text {for a. a. }x\in \Omega ,\text { for all }s\ge M \text { and for some }c_1>0. \end{aligned}$$

This implies, due to (2.3), that

$$\begin{aligned} c_1 s^{\mu -1} \le f(x,s)\quad \text {for a. a. }x\in \Omega \text { and for all }s\ge M. \end{aligned}$$

We see that the AR-condition is dictating that $f(x,\cdot )$ eventually has $(\mu -1)$-polynomial growth. Here, instead of the AR-condition, see (2.3), (2.4), we employ a less restrictive behavior near $+\infty$, see hypothesis H(iii). This way we are able to incorporate in our framework superlinear nonlinearities with “slower” growth near $+\infty$. For example, consider the function $f:\mathbb {R}\rightarrow \mathbb {R}$ (for the sake of simplicity we drop the x-dependence) defined by

$$\begin{aligned} f(x)= {\left\{ \begin{array}{ll} s^{\mu -1} &{}\text {if }0 \le s \le 1,\\ s^{p-1}\ln (x)+s^{\tilde{s}-1} &{}\text {if } 1<s \end{array}\right. } \end{aligned}$$

with $q<\mu <p$ and $\tilde{s}<p$, see (2.2). This function satisfies hypotheses H, but fails to satisfy the AR-condition.

By a solution of (P$_\lambda$) we mean a function $u\in W^{1,p}_0(\Omega )$, $u\ge 0$, $u\ne 0$, such that $uh\in L^{1}(\Omega )$ for all $h\in W^{1,p}_0(\Omega )$ and

$$\begin{aligned} \left\langle A_p(u),h\right\rangle +\left\langle A_q(u),h\right\rangle = \int _{\Omega }u^{-\eta }h\,\mathrm{d}x+\lambda \int _{\Omega }f(x,u)h\,\mathrm{d}x\quad \text {for all } h\in W^{1,p}_0(\Omega ). \end{aligned}$$

The energy functional $\varphi _\lambda :W^{1,p}_0(\Omega )\rightarrow \mathbb {R}$ of the problem (P$_\lambda$) is given by

$$\begin{aligned} \varphi _\lambda (u)=\frac{1}{p}\Vert \nabla u\Vert _p^p+\frac{1}{q}\Vert \nabla u\Vert _q^q -\frac{1}{1-\eta }\int _{\Omega }\left( u^+\right) ^{1-\eta }\,\mathrm{d}x -\lambda \int _{\Omega }F\left( x,u^+\right) \,\mathrm{d}x \end{aligned}$$

for all $h\in W^{1,p}_0(\Omega )$.

We can find solutions of (P$_\lambda$) among the critical points of $\varphi _\lambda$. The problem that we face is that because of the third term, so the singular one, the energy functional $\varphi _\lambda$ is not $C^1$. So, we cannot apply directly the minimax theorems of the critical point theory on $\varphi _\lambda$. Solving related auxiliary Dirichlet problems and then using suitable truncation and comparison techniques, we are able to overcome this difficulty, isolate the singularity and deal with $C^1$-functionals on which the classical critical point theory can be used.

To this end, first we consider the following purely singular Dirichlet problem

$$\begin{aligned} -\Delta _p u-\Delta _q u= & {} u^{-\eta }\quad \text {in } \Omega ,\nonumber \\ u= & {} 0 \quad \text {on } \partial \Omega ,\nonumber \\ u> & {} 0,\quad 0<\eta<1, 1<q<p. \end{aligned}$$

(2.5)

From Proposition 10 of Papageorgiou–Rădulescu–Repovš [12] we have the following result concerning problem (2.5).

Proposition 2.3

Problem (2.5) admits a unique solution $\underline{u}\in {{\,\mathrm{int}\,}}\left( C^1_0(\overline{\Omega })_+\right)$.

From the Lemma in Lazer-McKenna [9] we know that

$$\begin{aligned} \underline{u}^{-\eta } \in L^{1}(\Omega ). \end{aligned}$$

Moreover, from Hardy’s inequality we have

$$\begin{aligned} \underline{u}^{-\eta } h \in L^{1}(\Omega )\quad \text {and}\quad \int _{\Omega }\left| \underline{u}^{-\eta }h\right| \,\mathrm{d}x \le \hat{c} \Vert h\Vert \end{aligned}$$

for all $h \in W^{1,p}_0(\Omega )$. It follows that $\underline{u}^{-\eta }+1 \in W^{-1,p'}(\Omega )=W^{1,p}_0(\Omega )^*$.

So, we can consider a second auxiliary Dirichlet problem

$$\begin{aligned} \begin{aligned} -\Delta _p u-\Delta _q u&=\underline{u}^{-\eta }+1\quad&\text {in } \Omega ,\\ u&= 0&\text {on } \partial \Omega ,\\ 0&<\eta<1,\quad 1<q<p. \end{aligned} \end{aligned}$$

(2.6)

We show that (2.6) has a unique solution.

Proposition 2.4

Problem (2.6) admits a unique solution $\overline{u}\in {{\,\mathrm{int}\,}}\left( C^1_0(\overline{\Omega })_+\right)$.

Proof

Consider the operator $L:W^{1,p}_0(\Omega )\rightarrow W^{-1,p'}(\Omega )$ with $\frac{1}{p}+\frac{1}{p'}=1$ defined by

$$\begin{aligned} L(u)=A_p(u)+A_q(u) \quad \text {for all }u\in W^{1,p}_0(\Omega ). \end{aligned}$$

This operator is continuous, strictly monotone, hence maximal monotone and coercive. Since $\underline{u}^{-\eta }+1\in W^{-1,p'(\Omega )}$ (see the comments after Proposition 2.3), we can find $\overline{u} \in W^{1,p}_0(\Omega ), \overline{u}\ne 0$ such that

$$\begin{aligned} L\left( \overline{u}\right) =\underline{u}^{-\eta }+1. \end{aligned}$$

The strict monotonicity of L implies the uniqueness of $\overline{u}$ while Theorem B.1 of Giacomoni-Schindler-Takáč [7] implies that $\overline{u} \in C^1_0(\overline{\Omega })_+\setminus \{0\}$. Furthermore, we have

$$\begin{aligned} \Delta _p \overline{u}(x)+\Delta _q \overline{u}(x) \le 0\quad \text {for a.\,a.\,}x\in \Omega . \end{aligned}$$

Hence, from the nonlinear maximum principle, see Pucci-Serrin [17, pp. 111 and 120], we conclude that $\overline{u}\in {{\,\mathrm{int}\,}}\left( C^1_0(\overline{\Omega })_+\right)$. $\square$

3 Positive Solutions

We introduce the following two sets

$$\begin{aligned} \mathcal {L}&=\left\{ \lambda >0: \text {problem }{{{(P_\lambda )}}} {\text { has a positive solution}}\right\} ,\\ \mathcal {S}_\lambda&=\left\{ u: u\text { is a positive solution of problem }{{{(P_\lambda )}}}\right\} . \end{aligned}$$

Proposition 3.1

If hypotheses H hold, then $\mathcal {L}\ne \emptyset$.

Proof

Let $\overline{u}\in {{\,\mathrm{int}\,}}\left( C^1_0(\overline{\Omega })_+\right)$ be as in Proposition 2.4. Hypothesis H(i) implies that $f(\cdot ,\overline{u}(\cdot ))\in L^{\infty }(\Omega )$. So, we can find $\lambda _0>0$ such that

$$\begin{aligned} 0 \le \lambda _0 f\left(x,\overline{u}(x)\right) \le 1\quad \text {for a. a. }x\in \Omega . \end{aligned}$$

(3.1)

From the weak comparison principle (see Pucci-Serrin [17, Theorem 3.4.1, p. 61]), we have $\underline{u} \le \overline{u}$. So, for given $\lambda \in (0,\lambda _0]$, we can define the following truncation of the reaction of problem (P$_\lambda$)

$$\begin{aligned} g_\lambda (x,s)= {\left\{ \begin{array}{ll} \underline{u}(x)^{-\eta }+\lambda f(x,\underline{u}(x)) &{}\text {if }s<\underline{u}(x),\\ s^{-\eta }+\lambda f(x,s) &{}\text {if }\underline{u}(x) \le s \le \overline{u}(x),\\ \overline{u}(x)^{-\eta }+\lambda f(x,\overline{u}(x)) &{}\text {if }\overline{u}(x)<s. \end{array}\right. } \end{aligned}$$

(3.2)

This is a Carathéodory function. We set $G_\lambda (x,s)=\int _0^s g_\lambda (x,t)\,\mathrm{d}t$ and consider the $C^1$-functional $\psi _\lambda :W^{1,p}_0(\Omega )\rightarrow \mathbb {R}$ defined by

$$\begin{aligned} \psi _\lambda (u)= \frac{1}{p} \Vert \nabla u\Vert _p^p+\frac{1}{q}\Vert \nabla u\Vert _q^q -\int _{\Omega }G_\lambda (x,u)\,\mathrm{d}x\quad \text {for all }u \in W^{1,p}_0(\Omega ), \end{aligned}$$

see also Papageorgiou-Smyrlis [13, Proposition 3]. From (3.2) we see that $\psi _\lambda$ is coercive. Also, using the Sobolev embedding theorem, we see that $\psi _\lambda$ is sequentially weakly lower semicontinuous. So, by the Weierstraß-Tonelli theorem, we can find $u_\lambda \in W^{1,p}_0(\Omega )$ such that

$$\begin{aligned} \psi _\lambda (u_\lambda )=\min \left[\psi _\lambda (u)\,:\,u\in W^{1,p}_0(\Omega )\right]. \end{aligned}$$

This means, in particular, that $\psi _\lambda '(u_\lambda )=0$, which gives

$$\begin{aligned} \left\langle A_p(u_\lambda ),h\right\rangle +\left\langle A_q(u_\lambda ),h\right\rangle =\int _{\Omega }g_\lambda (x,u_\lambda )h\,\mathrm{d}x\quad \text {for all }h\in W^{1,p}_0(\Omega ). \end{aligned}$$

(3.3)

First, we choose $h=\left(\underline{u}-u_\lambda \right)^+\in W^{1,p}_0(\Omega )$ in (3.3). This yields, because of (3.2), $f \ge 0$ and Proposition 2.3 that

$$\begin{aligned}&\left\langle A_p(u_\lambda ),\left(\underline{u}-u_\lambda \right)^+\right\rangle +\left\langle A_q(u_\lambda ),\left(\underline{u}-u_\lambda \right)^+\right\rangle \\&\quad =\int _{\Omega }\left[\underline{u}^{-\eta }+\lambda f(x,\underline{u})\right] \left(\underline{u}-u_\lambda \right)^+\,\mathrm{d}x\\&\quad \ge \int _{\Omega }\underline{u}^{-\eta } \left(\underline{u}-u_\lambda \right)^+\,\mathrm{d}x\\&\quad =\left\langle A_p(\underline{u}),\left(\underline{u}-u_\lambda \right)^+\right\rangle +\left\langle A_q(\underline{u}),\left(\underline{u}-u_\lambda \right)^+\right\rangle . \end{aligned}$$

This implies

$$\begin{aligned}&\int _{\{\underline{u}>u_\lambda \}} \left(|\nabla \underline{u}|^{p-2} \nabla \underline{u} - |\nabla u_\lambda |^{p-2}\nabla u_\lambda \right) \cdot \left(\nabla \underline{u}-\nabla u_\lambda \right)\,\mathrm{d}x\\&\quad \quad +\int _{\{\underline{u}>u_\lambda \}} \left(|\nabla \underline{u}|^{q-2} \nabla \underline{u} - |\nabla u_\lambda |^{q-2}\nabla u_\lambda \right) \cdot \left(\nabla \underline{u}-\nabla u_\lambda \right)\,\mathrm{d}x\\&\quad \le 0, \end{aligned}$$

which means $|\{\underline{u}>u_\lambda \}|_N=0$ with $|\cdot |_N$ being the Lebesgue measure of $\mathbb {R}^N$. Hence,

$$\begin{aligned} \underline{u} \le u_\lambda . \end{aligned}$$

(3.4)

Next, we choose $h=\left(u_\lambda -\overline{u}\right)^+\in W^{1,p}_0(\Omega )$ in (3.3). Applying (3.2), (3.4), (3.1) and recall that $0 <\lambda \le \lambda _0$, we obtain

$$\begin{aligned}&\left\langle A_p(u_\lambda ),\left(u_\lambda -\overline{u}\right)^+\right\rangle +\left\langle A_q(u_\lambda ),\left(u_\lambda -\overline{u}\right)^+\right\rangle \\&\quad =\int _{\Omega }\left[\overline{u}^{-\eta }+\lambda f(x,\overline{u})\right] \left(u_\lambda -\overline{u}\right)^+\,\mathrm{d}x\\&\qquad \le \int _{\Omega }\left[ \underline{u}^{-\eta }+1\right]\left(u_\lambda -\overline{u}\right)^+\,\mathrm{d}x\\&\quad =\left\langle A_p(\overline{u}),\left(u_\lambda -\overline{u}\right)^+\right\rangle +\left\langle A_q(\overline{u}),\left(u_\lambda -\overline{u}\right)^+\right\rangle .\\ \end{aligned}$$

From this we see that

$$\begin{aligned}&\int _{\{u_\lambda>\overline{u}\}} \left(|\nabla u_\lambda |^{p-2} \nabla u_\lambda - |\nabla \overline{u}|^{p-2}\nabla \overline{u}\right) \cdot \left(\nabla u_\lambda -\nabla \overline{u}\right)\,\mathrm{d}x\\&\quad \quad +\int _{\{u_\lambda >\overline{u}\}} \left(|\nabla u_\lambda |^{q-2} \nabla u_\lambda - |\nabla \overline{u}|^{q-2}\nabla \overline{u}\right) \cdot \left(\nabla u_\lambda -\nabla \overline{u}\right)\,\mathrm{d}x\\&\quad \le 0 \end{aligned}$$

and so $|\{u_\lambda >\overline{u}\}|_N=0$. Thus, $u_\lambda \le \overline{u}$. So, we have proved that

$$\begin{aligned} u_\lambda \in [\underline{u},\overline{u}]. \end{aligned}$$

(3.5)

Then, (3.5), (3.2) and (3.3) imply that $u_\lambda \in \mathcal {S}_\lambda$ and so $(0,\lambda _0]\subseteq \mathcal {L}\ne \emptyset$. $\square$

Proposition 3.2

If hypotheses H hold and $\lambda \in \mathcal {L}$, then $\underline{u}\le u$ for all $u \in \mathcal {S}_\lambda$.

Proof

Let $u \in \mathcal {S}_\lambda$. On $\Omega \times (0,+\infty )$ we introduce the Carathéodory function $k(\cdot ,\cdot )$ defined by

$$\begin{aligned} k(x,s)= {\left\{ \begin{array}{ll} s^{-\eta } &{}\text {if }0<s\le u(x),\\ u(x)^{-\eta }&{}\text {if }u(x)<s \end{array}\right. } \end{aligned}$$

(3.6)

for all $(x,s)\in \Omega \times (0,+\infty )$. Then we consider the following Dirichlet (p, q)-problem

$$\begin{aligned} \begin{aligned} -\Delta _p u -\Delta _q u&=k(x,u) \quad&\text {in } \Omega ,\\ u&= 0&\text {on } \partial \Omega ,\\ u&>0, \quad 1<q<p. \end{aligned} \end{aligned}$$

Proposition 10 of Papageorgiou–Rădulescu–Repovš [12] implies that this problem admits a solution

$$\begin{aligned} \tilde{\underline{u}}\in {{\,\mathrm{int}\,}}\left( C^1_0(\overline{\Omega })_+\right) . \end{aligned}$$

(3.7)

This means

$$\begin{aligned} \left\langle A_p\left(\tilde{\underline{u}}\right),h\right\rangle +\left\langle A_q\left(\tilde{\underline{u}}\right),h\right\rangle =\int _{\Omega }k\left(x,\tilde{\underline{u}}\right)h\,\mathrm{d}x\quad \text {for all }h\in W^{1,p}_0(\Omega ). \end{aligned}$$

(3.8)

Choosing $h=\left(\tilde{\underline{u}}-u\right)^+\in W^{1,p}_0(\Omega )$ in (3.8) and applying (3.6), $f \ge 0$ and $u\in \mathcal {S}_\lambda$ gives

$$\begin{aligned}&\left\langle A_p(\tilde{\underline{u}}),\left(\tilde{\underline{u}}-u\right)^+\right\rangle +\left\langle A_q(\tilde{\underline{u}}),\left(\tilde{\underline{u}}-u\right)^+\right\rangle \\&\quad =\int _{\Omega }u^{-\eta } \left(\tilde{\underline{u}}-u\right)^+\,\mathrm{d}x\\&\qquad \le \int _{\Omega }\left[u^{-\eta }+\lambda f(x,u)\right]\left(\tilde{\underline{u}}-u\right)^+\,\mathrm{d}x\\&\quad =\left\langle A_p(u),\left(\tilde{\underline{u}}-u\right)^+\right\rangle +\left\langle A_q(u),\left(\tilde{\underline{u}}-u\right)^+\right\rangle . \end{aligned}$$

This implies

$$\begin{aligned}&\int _{\{\tilde{\underline{u}}>u\}} \left(|\nabla \tilde{\underline{u}}|^{p-2} \nabla \tilde{\underline{u}} - |\nabla u|^{p-2}\nabla u\right) \cdot \left(\nabla \tilde{\underline{u}}-\nabla u\right)\,\mathrm{d}x\\&\quad \quad +\int _{\{\tilde{\underline{u}}>u\}} \left(|\nabla \tilde{\underline{u}}|^{q-2} \nabla \tilde{\underline{u}} - |\nabla u|^{q-2}\nabla u\right) \cdot \left(\nabla \tilde{\underline{u}}-\nabla u\right)\,\mathrm{d}x\\&\quad \le 0, \end{aligned}$$

which means $|\{\tilde{\underline{u}}>u\}|_N=0$. Thus,

$$\begin{aligned} \tilde{\underline{u}} \le u. \end{aligned}$$

(3.9)

From (3.9), (3.7), (3.6), (3.8) and Proposition 2.3 it follows that $\tilde{\underline{u}}=u$. Therefore, $\underline{u} \le u$ for all $u \in \mathcal {S}_\lambda$. $\square$

As before, using Theorem B.1 of Giacomoni-Schindler-Takáč [7], we have the following result about the solution set $S_\lambda$.

Proposition 3.3

If hypotheses H hold and $\lambda \in \mathcal {L}$, then $S_\lambda \subseteq {{\,\mathrm{int}\,}}\left( C^1_0(\overline{\Omega })_+\right)$.

Let $\lambda ^*=\sup \mathcal {L}$.

Proposition 3.4

If hypotheses H hold, then $\lambda ^*<\infty$.

Proof

Hypotheses H(ii), (iii) imply that we can find $M>0$ such that

$$\begin{aligned} f(x,s) \ge s^{p-1}\quad \text {for a. a. }x\in \Omega \text { and for all }s \ge M. \end{aligned}$$

Moreover, hypothesis H(iv) implies that there exist $\delta \in (0,1)$ and $\hat{\eta }_1 \in (0,\hat{\eta })$ such that

$$\begin{aligned} f(x,s) \ge \hat{\eta }_1 s^{\tau -1}\ge \hat{\eta }_1 s^{p-1} \end{aligned}$$

for a. a. $x\in \Omega$ and for all $0\le s \le \delta$ since $\tau <p$ and $\delta <1$. This yields

$$\begin{aligned} \frac{1}{\hat{\eta }_1} f(x,s) \ge s^{p-1}\quad \text {for a. a. }x\in \Omega \text { and for all }0 \le s \le \delta . \end{aligned}$$

In addition, on account of hypothesis H(v) we can find $\tilde{\lambda }>0$ large enough such that

$$\begin{aligned} \tilde{\lambda } f(x,s) \ge M^{p-1} \quad \text {for a. a. }x\in \Omega \text { and for all } \delta \le s \le M. \end{aligned}$$

Therefore, taking into account the calculations above, there exists $\hat{\lambda }>0$ large enough such that

$$\begin{aligned} s^{p-1}\le \hat{\lambda }f(x,s)\quad \text {for a. a. }x\in \Omega \text { and for all }s \ge 0. \end{aligned}$$

(3.10)

Let $\lambda >\hat{\lambda }$ and suppose that $\lambda \in \mathcal {L}$. Then we can find $u_\lambda \in \mathcal {S}_\lambda \subseteq {{\,\mathrm{int}\,}}\left( C^1_0(\overline{\Omega })_+\right)$, see Proposition 3.3. Let $\Omega '\subset \subset \Omega$ with $C^2$-boundary $\partial \Omega '$. Then $m_0=\min _{\overline{\Omega '}} u_\lambda >0$ since $u_\lambda \in {{\,\mathrm{int}\,}}\left( C^1_0(\overline{\Omega })_+\right)$. Let $\rho =\Vert u_\lambda \Vert _\infty$ and let $\hat{\xi }_\rho >0$ be as postulated by hypothesis H(v). For $\delta >0$, we set $m_0^\delta =m_0+\delta$. Applying (3.10), hypothesis H(v) and $u_\lambda \in \mathcal {S}_\lambda$, we have for a. a. $x\in \Omega '$

$$\begin{aligned}&-\Delta _p m_0^\delta -\Delta _q m_0^\delta +\lambda \hat{\xi }_\rho \left(m_0^\delta \right)^{p-1}-\lambda \left(m_0^\delta \right)^{-\eta }\\&\quad \le \lambda \hat{\xi }_\rho m_0^{p-1}+\chi (\delta ) \quad \text {with }\chi (\delta )\rightarrow 0^+ \text { as }\delta \rightarrow 0^+\\&\quad \le \left[\lambda \hat{\xi }_\rho +1\right]m_0^{p-1}+\chi (\delta )\\&\quad \le \hat{\lambda }f(x,m_0)+\lambda \hat{\xi }_\rho m_0^{p-1}+\chi (\delta )\\&\quad = \lambda \left[f(x,m_0)+\hat{\xi }_\rho m_0^{p-1}\right]-\left(\lambda -\hat{\lambda }\right)f(x,m_0) +\chi (\delta )\\&\quad \le \lambda \left[ f\left(x,u_\lambda (x)\right)+\hat{\xi }_\rho u_\lambda (x)^{p-1}\right] \quad \text {for }\delta >0\text { small enough}\\&\quad =-\Delta _p u_\lambda (x)-\Delta _q u_\lambda (x) +\lambda \hat{\xi }_\rho u_\lambda (x)^{p-1}-\lambda u_\lambda (x)^{-\eta }. \end{aligned}$$

Note that for $\delta >0$ small enough, we will have

$$\begin{aligned} 0<\hat{\eta } \le \left[\lambda -\hat{\lambda }\right]f(x,m_0)-\chi (\delta )\quad \text {for a.\,a.\,}x\in \Omega ', \end{aligned}$$

see hypothesis H(v). Then, invoking Proposition 6 of Papageorgiou–Rădulescu–Repovš [12], it follows that

$$\begin{aligned} m_0^\delta <u_\lambda (x)\quad \text {for a.\,a.\,}x\in \Omega '\text { and for }\delta >0 \text { small enough}, \end{aligned}$$

which contradicts the definition of $m_0$. Therefore, $\lambda \not \in \mathcal {L}$ and so we conclude that $\lambda ^*\le \hat{\lambda }<\infty$. $\square$

Next, we are going to show that $\mathcal {L}$ is an interval. So, we have

$$\begin{aligned} \left(0,\lambda ^*\right)\subseteq \mathcal {L}\subseteq \left(0,\lambda ^*\right]. \end{aligned}$$

Proposition 3.5

If hypotheses H hold, $\lambda \in \mathcal {L}$ and $0<\mu <\lambda$, then $\mu \in \mathcal {L}$.

Proof

Since $\lambda \in \mathcal {L}$, we can find $u_\lambda \in \mathcal {S}_\lambda \subseteq {{\,\mathrm{int}\,}}\left( C^1_0(\overline{\Omega })_+\right)$. We know that $\underline{u}\le u_\lambda$, see Proposition 3.2. So, we can define the following truncation $e_\mu :\Omega \times \mathbb {R}\rightarrow \mathbb {R}$ of the reaction for problem (P$_\lambda$)

$$\begin{aligned} e_\mu (x,s)= {\left\{ \begin{array}{ll} \underline{u}(x)^{-\eta }+\mu f(x,\underline{u}(x))&{}\text {if }s<\underline{u}(x),\\ s^{-\eta }+\mu f(x,s)&{}\text {if }\underline{u}(x)\le s\le u_\lambda (x),\\ u_\lambda (x)^{-\eta }+\mu f\left(x,u_\lambda (x)\right)&{}\text {if }u_\lambda (x)<s, \end{array}\right. } \end{aligned}$$

(3.11)

which is a Carathéodory function. We set $E_\mu (x,s)=\int ^s_0 e_\mu (x,t)\,\mathrm{d}t$ and consider the $C^1$-functional $\hat{\varphi }_\mu :W^{1,p}_0(\Omega )\rightarrow \mathbb {R}$ defined by

$$\begin{aligned} \hat{\varphi }_\mu (u)=\frac{1}{p}\Vert \nabla u\Vert _p^p+\frac{1}{q}\Vert \nabla u\Vert _q^q -\int _{\Omega }E_\mu (x,u)\,\mathrm{d}x\quad \text {for all }u\in W^{1,p}_0(\Omega ), \end{aligned}$$

see Papageorgiou-Vetro-Vetro [14]. From (3.11) it is clear that $\hat{\varphi }_\mu$ is coercive. Moreover, it is sequentially weakly lower semicontinuous. Therefore, we can find $u_\mu \in W^{1,p}_0(\Omega )$ such that

$$\begin{aligned} \hat{\varphi }_\mu \left(u_\mu \right)= \min \left[\hat{\varphi }_\mu (u)\,:\,u\in W^{1,p}_0(\Omega )\right]. \end{aligned}$$

In particular, we have $\hat{\varphi }_\mu '\left(u_\mu \right)=0$ which means

$$\begin{aligned} \left\langle A_p\left(u_\mu \right),h\right\rangle +\left\langle A_q\left(u_\mu \right),h\right\rangle =\int _{\Omega }e_\mu (x,u)h\,\mathrm{d}x\quad \text {for all }h\in W^{1,p}_0(\Omega ). \end{aligned}$$

(3.12)

Choosing $h=\left(\underline{u}-u_\mu \right)^+\in W^{1,p}_0(\Omega )$ in (3.12) and applying (3.11), $f\ge 0$ and Proposition 2.3 yields

$$\begin{aligned}&\left\langle A_p\left(u_\mu \right),\left(\underline{u}-u_\mu \right)^+\right\rangle +\left\langle A_q\left(u_\mu \right),\left(\underline{u}-u_\mu \right)^+\right\rangle \\&\quad =\int _{\Omega }\left[\underline{u}^{-\eta }+\mu f(x,\underline{u})\right]\left(\underline{u}-u_\mu \right)^+\,\mathrm{d}x\\&\qquad \ge \int _{\Omega }\underline{u}^{-\eta } \left(\underline{u}-u_\mu \right)^+\,\mathrm{d}x\\&\quad =\left\langle A_p\left(\underline{u}\right),\left(\underline{u}-u_\mu \right)^+\right\rangle +\left\langle A_q\left(\underline{u}\right),\left(\underline{u}-u_\mu \right)^+\right\rangle . \end{aligned}$$

We obtain $\underline{u} \le u_\mu$. Furthermore, choosing $h=\left(u_\mu -u_\lambda \right)^+\in W^{1,p}_0(\Omega )$ in (3.12) and applying (3.11), $\mu <\lambda$ and $u_\lambda \in \mathcal {S}_\lambda$, we get

$$\begin{aligned}&\left\langle A_p\left(u_\mu \right),\left(u_\mu -u_\lambda \right)^+\right\rangle +\left\langle A_q\left(u_\mu \right),\left(u_\mu -u_\lambda \right)^+\right\rangle \\&\quad =\int _{\Omega }\left[u_\lambda ^{-\eta }+\mu f(x,u_\lambda )\right]\left(u_\mu -u_\lambda \right)^+\,\mathrm{d}x\\&\qquad \le \int _{\Omega }\left[u^{-\eta }+\lambda f(x,u_\lambda )\right]\left(u_\mu -u_\lambda \right)^+\,\mathrm{d}x\\&\quad =\left\langle A_p\left(u_\lambda \right),\left(u_\mu -u_\lambda \right)^+\right\rangle +\left\langle A_q\left(u_\lambda \right),\left(u_\mu -u_\lambda \right)^+\right\rangle . \end{aligned}$$

Hence, $u_\mu \le u_\lambda$ and so we have proved that

$$\begin{aligned} u_\mu \in \left[\underline{u},u_\lambda \right]. \end{aligned}$$

(3.13)

From (3.13), (3.11) and (3.12) we infer that

$$\begin{aligned} u_\mu \in \mathcal {S}_\mu \subseteq {{\,\mathrm{int}\,}}\left( C^1_0(\overline{\Omega })_+\right) . \end{aligned}$$

Thus, $\mu \in \mathcal {L}$. $\square$

A byproduct of the proof above is the following corollary.

Corollary 3.6

If hypotheses H hold, $\lambda \in \mathcal {L}$, $u_\lambda \in \mathcal {S}_\lambda \subseteq {{\,\mathrm{int}\,}}\left( C^1_0(\overline{\Omega })_+\right)$ and $\mu \in (0,\lambda )$, then $\mu \in \mathcal {L}$ and there exists $u_\mu \in \mathcal {S}_\mu \subseteq {{\,\mathrm{int}\,}}\left( C^1_0(\overline{\Omega })_+\right)$ such that $u_\mu \le u_\lambda$.

Using the strong comparison principle of Papageorgiou–Rădulescu–Repovš [12] we can improve the conclusion of this corollary as follows.

Proposition 3.7

If hypotheses H hold, $\lambda \in \mathcal {L}$, $u_\lambda \in \mathcal {S}_\lambda \subseteq {{\,\mathrm{int}\,}}\left( C^1_0(\overline{\Omega })_+\right)$ and $\mu \in (0,\lambda )$, then $\mu \in \mathcal {L}$ and there exists $u_\mu \in \mathcal {S}_\mu \subseteq {{\,\mathrm{int}\,}}\left( C^1_0(\overline{\Omega })_+\right)$ such that

$$\begin{aligned} u_\lambda -u_\mu \in {{\,\mathrm{int}\,}}\left( C^1_0(\overline{\Omega })_+\right) . \end{aligned}$$

Proof

From Corollary 3.6 we already have that $\mu \in \mathcal {L}$ and we also know that there exists $u_\mu \in \mathcal {S}_\mu \subseteq {{\,\mathrm{int}\,}}\left( C^1_0(\overline{\Omega })_+\right)$ such that

$$\begin{aligned} u_\mu \le u_\lambda . \end{aligned}$$

(3.14)

Let $\rho =\Vert u_\lambda \Vert _\infty$ and let $\hat{\xi }_\rho >0$ be as postulated by hypothesis H(v). Applying $u_\mu \in \mathcal {S}_\mu$, (3.14), hypothesis H(v) and $\mu <\lambda$, we obtain

$$\begin{aligned}&-\Delta _p u_\mu (x)-\Delta _q u_\mu (x)+\lambda \hat{\xi }_\rho u_\mu (x)^{p-1}-u_\mu (x)^{-\eta }\nonumber \\&\quad =\mu f(x,u_\mu (x))+\lambda \hat{\xi }_\rho u_\mu (x)^{p-1}\nonumber \\&\quad =\lambda \left[f(x,u_\mu (x))+\hat{\xi }_\rho u_\mu (x)^{p-1}\right]-(\lambda -\mu )f(x,u_\mu (x))\nonumber \\&\qquad \le \lambda \left[f(x,u_\lambda (x)) +\hat{\xi }_\rho u_\lambda (x)^{p-1}\right] \nonumber \\&\quad =-\Delta _pu_\lambda (x)-\Delta _q u_\lambda (x) +\lambda \hat{\xi }_\rho u_\lambda (x)^{p-1}-u_\lambda (x)^{-\eta } \end{aligned}$$

(3.15)

for a. a. $x\in \Omega$. Since $u_\mu \in {{\,\mathrm{int}\,}}\left( C^1_0(\overline{\Omega })_+\right)$, because of hypothesis H(v), we have

$$\begin{aligned} 0\prec (\lambda -\mu )f(\cdot ,u_\mu (\cdot )). \end{aligned}$$

Then, from (3.15) and Proposition 7 of Papageorgiou–Rădulescu–Repovš [12] we conclude that $u_\lambda -u_\mu \in {{\,\mathrm{int}\,}}\left( C^1_0(\overline{\Omega })_+\right)$. $\square$

Proposition 3.8

If hypotheses H hold and $\lambda \in (0,\lambda ^*)$, then problem (P$_\lambda$) has at least two positive solutions

$$\begin{aligned} u_0, \hat{u} \in {{\,\mathrm{int}\,}}\left( C^1_0(\overline{\Omega })_+\right) ,\quad u_0\le \hat{u}, \quad u_0 \ne \hat{u}. \end{aligned}$$

Proof

Let $\lambda<\vartheta <\lambda ^*$. Due to Proposition 3.7, we can find $u_\vartheta \in \mathcal {S}_\vartheta \subseteq {{\,\mathrm{int}\,}}\left( C^1_0(\overline{\Omega })_+\right)$ and $u_0\in \mathcal {S}_\lambda$ such that

$$\begin{aligned} u_\vartheta -u_0\in {{\,\mathrm{int}\,}}\left( C^1_0(\overline{\Omega })_+\right) . \end{aligned}$$

(3.16)

From Proposition 3.2 we know that $\underline{u}\le u_0$. Therefore, $u_0^{-\eta } \in L^{1}(\Omega )$. So, we can define the following truncation $w_\lambda :\Omega \times \mathbb {R}\rightarrow \mathbb {R}$ of the reaction in problem (P$_\lambda$)

$$\begin{aligned} w_\lambda (x,s)= {\left\{ \begin{array}{ll} u_0(x)^{-\eta }+\lambda f(x,u_0(x))&{}\text {if }s\le u_0(x),\\ s^{-\eta }+\lambda f(x,s)&{}\text {if }u_0(x)< s. \end{array}\right. } \end{aligned}$$

(3.17)

Also, using (3.16), we can consider the truncation $\hat{w}_\lambda :\Omega \times \mathbb {R}\rightarrow \mathbb {R}$ of $w_\lambda (x,\cdot )$ defined by

$$\begin{aligned} \hat{w}_\lambda (x,s)= {\left\{ \begin{array}{ll} w_\lambda (x,s)&{}\text {if }s\le u_\vartheta (x),\\ w_\lambda (x,u_\vartheta (x))&{}\text {if }u_\vartheta (x)< s. \end{array}\right. } \end{aligned}$$

(3.18)

It is clear that both are Carathéodory function. We set

$$\begin{aligned} W_\lambda (x,s)=\int ^s_0 w_\lambda (x,t)\,\mathrm{d}t\quad \text {and}\quad \hat{W}_\lambda (x,s)=\int ^s_0 \hat{w}_\lambda (x,t)\,\mathrm{d}t \end{aligned}$$

and consider the $C^1$-functionals $\sigma _\lambda , \hat{\sigma }_\lambda :W^{1,p}_0(\Omega )\rightarrow \mathbb {R}$ defined by

$$\begin{aligned} \sigma _\lambda (u)&=\frac{1}{p}\Vert \nabla u\Vert _p^p +\frac{1}{q}\Vert \nabla u\Vert _q^q-\int _{\Omega }W_\lambda (x,u)\,\mathrm{d}x\quad \text {for all }u\in W^{1,p}_0(\Omega ),\\ \hat{\sigma }_\lambda (u)&=\frac{1}{p}\Vert \nabla u\Vert _p^p +\frac{1}{q}\Vert \nabla u\Vert _q^q-\int _{\Omega }\hat{W}_\lambda (x,u)\,\mathrm{d}x \quad \text {for all }u\in W^{1,p}_0(\Omega ). \end{aligned}$$

From (3.17) and (3.18) it is clear that

$$\begin{aligned} \sigma _\lambda \big |_{[0,u_\vartheta ]}=\hat{\sigma }_\lambda \big |_{[0,u_\vartheta ]} \quad \text {and}\quad \sigma '_\lambda \big |_{[0,u_\vartheta ]}=\hat{\sigma }'_\lambda \big |_{[0,u_\vartheta ]}. \end{aligned}$$

(3.19)

Using (3.17), (3.18) and the nonlinear regularity theory of Lieberman [10] we obtain that

$$\begin{aligned} K_{\sigma _\lambda } \subseteq [u_0)\cap {{\,\mathrm{int}\,}}\left( C^1_0(\overline{\Omega })_+\right) \quad \text {and}\quad K_{\hat{\sigma }_\lambda }\subseteq [u_0,u_\vartheta ]\cap {{\,\mathrm{int}\,}}\left( C^1_0(\overline{\Omega })_+\right) . \end{aligned}$$

(3.20)

From (3.20) we see that we may assume that

$$\begin{aligned} K_{\sigma _\lambda } \text { is finite and } K_{\sigma _\lambda } \cap [u_0,u_\vartheta ]=\{u_0\}. \end{aligned}$$

(3.21)

Otherwise we already have a second positive smooth solution larger that $u_0$ and so we are done.

From (3.18) and since $u_0^{-\eta } \in L^{1}(\Omega )$, it is clear that $\hat{\sigma }_\lambda$ is coercive and it is also sequentially weakly lower semicontinuous. Hence, we find its global minimizer $\tilde{u}_0 \in W^{1,p}_0(\Omega )$ such that

$$\begin{aligned} \hat{\sigma }_\lambda \left(\tilde{u}_0\right)=\min \left[\hat{\sigma }_\lambda (u)\,:\,u\in W^{1,p}_0(\Omega )\right]. \end{aligned}$$

By (3.20) we see that $\tilde{u}_0\in K_{\hat{\sigma }_\lambda }\subseteq [u_0,u_\vartheta ]\cap {{\,\mathrm{int}\,}}\left( C^1_0(\overline{\Omega })_+\right)$. Then, (3.19) and (3.21) imply $\tilde{u}_0=u_0\in {{\,\mathrm{int}\,}}\left( C^1_0(\overline{\Omega })_+\right)$. Finally, from (3.16) we obtain that $u_0$ is a local $C^1_0(\overline{\Omega })$-minimizer of $\sigma _\lambda$ and then by Gasiński-Papageorgiou [6] we have that

$$\begin{aligned} u_0 \text { is also a local }W^{1,p}_0(\Omega )\text {-minimizer of }\sigma _\lambda . \end{aligned}$$

(3.22)

From (3.22), (3.21) and Theorem 5.7.6 of Papageorgiou–Rădulescu–Repovš [11, p. 449] we know that we can find $\rho \in (0,1)$ small enough such that

$$\begin{aligned} \sigma _\lambda (u_0)<\inf \left[\sigma _\lambda (u):\Vert u-u_0\Vert =\rho \right]=m_\lambda . \end{aligned}$$

(3.23)

Hypothesis H(ii) implies that if $u\in {{\,\mathrm{int}\,}}\left( C^1_0(\overline{\Omega })_+\right)$, then

$$\begin{aligned} \sigma _\lambda (tu)\rightarrow -\infty \quad \text {as }t\rightarrow +\infty . \end{aligned}$$

(3.24)

Claim: The functional $\sigma _\lambda$ satisfies the C-condition.

Consider a sequence $\{u_n\}_{n\ge 1}\subseteq W^{1,p}_0(\Omega )$ such that

$$\begin{aligned} |\sigma _\lambda (u_n)|\le c_6\quad \text {for some }c_6>0 \text { and for all }n\in \mathbb {N}, \end{aligned}$$

(3.25)

$$\begin{aligned} (1+\Vert u_n\Vert )\sigma '_\lambda (u_n) \rightarrow 0 \text { in } W^{-1,p'}(\Omega ) \text { as }n\rightarrow \infty . \end{aligned}$$

(3.26)

From (3.26) we have

$$\begin{aligned} \left|\left\langle A_p(u_n),h\right\rangle +\left\langle A_q(u_n),h\right\rangle - \int _{\Omega }w_\lambda (x,u_n)h\,\mathrm{d}x \right| \le \frac{\varepsilon _n \Vert h\Vert }{1+\Vert u_n\Vert } \end{aligned}$$

(3.27)

for all $h\in W^{1,p}_0(\Omega )$ with $\varepsilon _n\rightarrow 0^+$. We choose $h=-u_n^-\in W^{1,p}_0(\Omega )$ in (3.27) and obtain, by applying (3.17), that

$$\begin{aligned} \Vert u_n^-\Vert ^p \le c_7 \quad \text {for some }c_7>0 \text { and for all }n\in \mathbb {N}. \end{aligned}$$

This shows that

$$\begin{aligned} \left\rbrace u_n^-\right\lbrace _{n\ge 1} \subseteq W^{1,p}_0(\Omega ) \text { is bounded}. \end{aligned}$$

(3.28)

From (3.25) and (3.28) it follows that

$$\begin{aligned} \Vert \nabla u_n^+\Vert _p^p +\frac{p}{q}\Vert \nabla u_n^+\Vert _q^q-\int _{\Omega }pF\left(x,u_n^+\right)\,\mathrm{d}x \le c_8\left[1+\Vert u_n^+\Vert _\tau \right] \end{aligned}$$

(3.29)

for some $c_8>0$ and for all $n \in \mathbb {N}$, see (3.17). Moreover, choosing $h=u_n^+\in W^{1,p}_0(\Omega )$ in (3.27), we obtain using (3.17)

$$\begin{aligned} -\Vert \nabla u_n^+\Vert _p^p-\Vert \nabla u_n^+\Vert _q^q +\int _{\Omega }f\left(x,u_n^+\right)u_n^+\,\mathrm{d}x \le c_9 \end{aligned}$$

(3.30)

for some $c_9>0$ and for all $n\in \mathbb {N}$. Adding (3.29) and (3.30) and recall that $q<p$, gives

$$\begin{aligned} \int _{\Omega }\big [ f\left(x,u_n^+\right)u_n^+-pF\left(x,u_n^+\right)\big ]\,\mathrm{d}x\le c_{10}\left[1+\Vert u_n^+\Vert _\tau \right] \end{aligned}$$

(3.31)

for some $c_{10}>0$ and for all $n\in \mathbb {N}$.

Taking hypotheses H(i), (iii) into account, we see that we can find constants $c_{11}, c_{12}>0$ such that

$$\begin{aligned} c_{11} s^\tau -c_{12} \le f(x,s)s-pF(x,s)\quad \text {for a.\,a.\,}x\in \Omega \text { and for all }s\ge 0. \end{aligned}$$

(3.32)

Applying (3.32) in (3.31), we infer that

$$\begin{aligned} \Vert u_n^+\Vert _\tau ^{\tau -1} \le c_{13} \end{aligned}$$

for some $c_{13}>0$ and for all $n\in \mathbb {N}$. Therefore,

$$\begin{aligned} \left\rbrace u_n^+\right\lbrace _{n\ge 1} \subseteq L^{\tau }(\Omega ) \text { is bounded}. \end{aligned}$$

(3.33)

First assume that $p\ne N$. From hypothesis H(iii), we see that we can always assume that $\tau<r<p^*$. So, we can find $t\in (0,1)$ such that

$$\begin{aligned} \frac{1}{r}=\frac{1-t}{\tau }+\frac{t}{p^*}. \end{aligned}$$

(3.34)

Invoking the interpolation inequality, see Papageorgiou-Winkert [16, Proposition 2.3.17, p. 116], we have

$$\begin{aligned} \Vert u_n^+\Vert _r \le \Vert u_n^+\Vert _\tau ^{1-r} \Vert u_n^+\Vert ^t_{p^*}. \end{aligned}$$

Hence, by (3.33),

$$\begin{aligned} \Vert u_n^+\Vert _r^r \le c_{14} \Vert u_n^+\Vert ^{tr} \end{aligned}$$

(3.35)

for some $c_{14}>0$ and for all $n\in \mathbb {N}$. We choose $h=u_n^+\in W^{1,p}_0(\Omega )$ in (3.27) to get

$$\begin{aligned} \Vert u_n^+\Vert ^p \le \int _{\Omega }w_\lambda \left(x,u_n^+\right)u_n^+\,\mathrm{d}x. \end{aligned}$$

Then, from (3.17) and hypothesis H(i), it follows that

$$\begin{aligned} \Vert u_n^+\Vert ^p \le \int _{\Omega }c_{15} \left[1+\left(u_n^+\right)^r\right]\,\mathrm{d}x \end{aligned}$$

for some $c_{15}>0$ and for all $n\in \mathbb {N}$. This implies

$$\begin{aligned} \Vert u_n^+\Vert ^p \le c_{16} \left[1+\Vert u_n^+\Vert _r^r\right] \end{aligned}$$

for some $c_{16}>0$ and for all $n\in \mathbb {N}$. Finally, from (3.35), we then obtain

$$\begin{aligned} \Vert u_n^+\Vert ^p \le c_{17} \left[1+\Vert u_n^+\Vert ^{tr}\right] \end{aligned}$$

(3.36)

for some $c_{17}>0$ and for all $n\in \mathbb {N}$.

If $N<p$, then $p^*=\infty$ and so from (3.34) we have $tr=r-\tau$, which by hypothesis H(iii) leads to $tr<p$.

If $N>p$, then $p^*=\frac{Np}{N-p}$. From (3.34) it follows

$$\begin{aligned} tr=\frac{(r-\tau )p^*}{p^*-\tau }, \end{aligned}$$

which implies

$$\begin{aligned} tr=\frac{(r-\tau )Np}{N(p-\tau )+\tau p}<p. \end{aligned}$$

Therefore, from (3.36) we infer that

$$\begin{aligned} \left\rbrace u_n^+\right\lbrace _{n\ge 1} \subseteq W^{1,p}_0(\Omega )\text { is bounded.} \end{aligned}$$

(3.37)

If $N=p$, then by the Sobolev embedding theorem, we know that $W^{1,p}_0(\Omega )\hookrightarrow L^{s}(\Omega )$ continuously for all $1\le s<\infty$. So, for the argument above to work, we need to replace $p^*$ by $s>r>\tau$ in (3.34) which yields

$$\begin{aligned} \frac{1}{r}=\frac{1-t}{\tau }+\frac{t}{s}. \end{aligned}$$

Then, by hypothesis H(iii), we obtain

$$\begin{aligned} tr=\frac{(r-\tau )s}{s-\tau }\rightarrow r-\tau <p \quad \text {as }s\rightarrow +\infty . \end{aligned}$$

We choose $s>r$ large enough so that $tr<p$. Then, we reach again (3.37).

From (3.37) and (3.28) it follows that

$$\begin{aligned} \left\rbrace u_n\right\lbrace _{n\ge 1}\subseteq W^{1,p}_0(\Omega )\text { is bounded}. \end{aligned}$$

So, we may assume that

$$\begin{aligned} u_n\rightharpoonup u \text { in }W^{1,p}_0(\Omega )\quad \text {and}\quad u_n\rightarrow u \text { in }L^{r}(\Omega ). \end{aligned}$$

(3.38)

In (3.27) we choose $h=u_n-u\in W^{1,p}_0(\Omega )$, pass to the limit as $n\rightarrow \infty$ and use (3.38). This gives

$$\begin{aligned} \lim _{n\rightarrow \infty } \left[\left\langle A_p(u_n),u_n-u\right\rangle +\left\langle A_q(u_n),u_n-u\right\rangle \right] =0. \end{aligned}$$

The monotonicity of $A_q$ implies

$$\begin{aligned} \lim _{n\rightarrow \infty } \left[\left\langle A_p(u_n),u_n-u\right\rangle +\left\langle A_q(u),u_n-u\right\rangle \right] \le 0 \end{aligned}$$

and from (3.38) one has

$$\begin{aligned} \limsup _{n\rightarrow \infty } \left\langle A_p(u_n),u_n-u\right\rangle \le 0. \end{aligned}$$

Hence, by Proposition 2.1, it follows

$$\begin{aligned} u_n\rightarrow u \quad \text {in }W^{1,p}_0(\Omega ). \end{aligned}$$

Therefore, $\sigma _\lambda$ satisfies the C-condition and this proves the Claim.

Then, (3.23), (3.24) and the Claim permit the use of the mountain pass theorem. So, we can find $\hat{u}\in W^{1,p}_0(\Omega )$ such that

$$\begin{aligned} \hat{u} \in K_{\sigma _\lambda }\subseteq [u_0)\cap {{\,\mathrm{int}\,}}\left( C^1_0(\overline{\Omega })_+\right) \quad \text {and}\quad \sigma _\lambda (u_0)<m_\lambda \le \sigma _\lambda \left(\hat{u}\right), \end{aligned}$$

(3.39)

see (3.20) and (3.23), respectively.

From (3.39), (3.17) and (3.27), we conclude that

$$\begin{aligned} \hat{u} \in \mathcal {S}_\lambda \subseteq {{\,\mathrm{int}\,}}\left( C^1_0(\overline{\Omega })_+\right) ,\quad u_0\le \hat{u}, \quad u_0 \ne \hat{u}. \end{aligned}$$

$\square$

Proposition 3.9

If hypotheses H hold, then $\lambda ^*\in \mathcal {L}$.

Proof

Let $0<\lambda _n <\lambda ^*$ with $n\in \mathbb {N}$ and assume that $\lambda _n\nearrow \lambda ^*$. By Proposition 3.2 we can find $u_n\in \mathcal {S}_{\lambda _n}\subseteq {{\,\mathrm{int}\,}}\left( C^1_0(\overline{\Omega })_+\right)$ such that

$$\begin{aligned} \underline{u}\le u_n\quad \text {for all }n\in \mathbb {N}\end{aligned}$$

and

$$\begin{aligned} \left\langle A_p(u_n),h\right\rangle +\left\langle A_q(u_n),h\right\rangle =\int _{\Omega }\left[ u_n^{-\eta }+\lambda _n f(x,u_n)\right]h\,\mathrm{d}x \end{aligned}$$

(3.40)

for all $h\in W^{1,p}_0(\Omega )$ and for all $n\in \mathbb {N}$. From hypothesis H(iii), we have

$$\begin{aligned} \varphi _\lambda (u_n) \le c_{18} \end{aligned}$$

(3.41)

for some $c_{18}>0$ and for all $n\in \mathbb {N}$, where $\varphi _\lambda$ is the energy functional of problem (P$_\lambda$).

From (3.40), (3.41) and reasoning as in the Claim in the proof of Proposition 3.8, we obtain that

$$\begin{aligned} u_n\rightarrow u_*\quad \text {in }W^{1,p}_0(\Omega ). \end{aligned}$$

(3.42)

So, if in (3.40) we pass to the limit as $n\rightarrow \infty$ and use (3.42), then

$$\begin{aligned} \left\langle A_p(u_*),h\right\rangle +\left\langle A_q(u_*),h\right\rangle =\int _{\Omega }\left[ u_*^{-\eta }+\lambda ^* f(x,u_*)\right]h\,\mathrm{d}x \end{aligned}$$

for all $h\in W^{1,p}_0(\Omega )$ and $\underline{u}\le u_*$. It follows that $u_*\in \mathcal {S}_{\lambda ^*}\subseteq {{\,\mathrm{int}\,}}\left( C^1_0(\overline{\Omega })_+\right)$ and so $\lambda ^*\in \mathcal {L}$. $\square$

Therefore, we have

$$\begin{aligned} \mathcal {L}=\left(0,\lambda ^*\right]. \end{aligned}$$

We can state the following bifurcation-type theorem describing the variations in the set of positive solutions as the parameter $\lambda$ moves in $(0,+\infty )$.

Theorem 3.10

If hypotheses H hold, then there exist $\lambda ^*>0$ such that

• (a) for every $0<\lambda <\lambda ^*$, problem (P$_\lambda$) has at least two positive solutions

  $$\begin{aligned} u_0, \hat{u} \in {{\,\mathrm{int}\,}}\left( C^1_0(\overline{\Omega })_+\right) , \quad u_0 \le \hat{u},\quad u_0\ne \hat{u}; \end{aligned}$$

• (b) for $\lambda =\lambda ^*$, problem (P$_\lambda$) has at least one positive solution

  $$\begin{aligned} u_*\in {{\,\mathrm{int}\,}}\left( C^1_0(\overline{\Omega })_+\right) ; \end{aligned}$$

• (c) for every $\lambda >\lambda ^*$, problem (P$_\lambda$) has no positive solutions.

4 Minimal Positive Solutions

In this section we show that for every $\lambda \in \mathcal {L}=(0,\lambda ^*]$, problem (P$_\lambda$) has a smallest positive solutions $u^*\in {{\,\mathrm{int}\,}}\left( C^1_0(\overline{\Omega })_+\right)$ and we investigate the monotonicity and continuity properties of the map $\lambda \rightarrow u^*_\lambda$.

Proposition 4.1

If hypotheses H hold and $\lambda \in \mathcal {L}$, then problem (P$_\lambda$) has a smallest positive solution $u^*_\lambda \in \mathcal {S}_\lambda \subseteq {{\,\mathrm{int}\,}}\left( C^1_0(\overline{\Omega })_+\right)$, that is, $u^*_\lambda \le u$ for all $u \in \mathcal {S}_\lambda$.

Proof

From Proposition 18 of Papageorgiou–Rădulescu–Repovš [12] we know that the set $\mathcal {S}_\lambda \subseteq W^{1,p}_0(\Omega )$ is downward directed. So, invoking Lemma 3.10 of Hu-Papageorgiou [8, p. 178], we can find a decreasing sequence $\{u_n\}_{n\ge 1}\subseteq \mathcal {S}_\lambda$ such that

$$\begin{aligned} \underline{u} \le u_n\le u_1 \text { for all }n\in \mathbb {N},\quad \inf _{n\ge 1} u_n = \inf \mathcal {S}_\lambda , \end{aligned}$$

(4.1)

see Proposition 3.2. From (4.1) we see that $\{u_n\}_{n\ge 1}\subseteq W^{1,p}_0(\Omega )$ is bounded. From this, as in the proof of Proposition 3.8, using Proposition 2.1, we obtain

$$\begin{aligned} u_n \rightarrow u^*_\lambda \quad \text {in }W^{1,p}_0(\Omega ), \quad \underline{u}\le u^*_\lambda . \end{aligned}$$

From (4.1) it follows

$$\begin{aligned} u^*_\lambda \in \mathcal {S}_\lambda \subseteq {{\,\mathrm{int}\,}}\left( C^1_0(\overline{\Omega })_+\right) \quad \text {and}\quad u^*_\lambda =\inf \mathcal {S}_\lambda . \end{aligned}$$

$\square$

In the next proposition we examine the monotonicity and continuity properties of the map $\lambda \rightarrow u^*_\lambda$ from $\mathcal {L}=(0,\lambda ^*]$ into $C^1_0(\overline{\Omega })$.

Proposition 4.2

If hypotheses H hold, then the minimal solution map $\lambda \rightarrow u^*_\lambda$ from $\mathcal {L}=(0,\lambda ^*]$ into $C^1_0(\overline{\Omega })$ is

• (a) strictly increasing in the sense that

  $$\begin{aligned} 0<\mu <\lambda \le \lambda ^* \quad \text {implies}\quad u^*_\lambda -u^*_\mu \in {{\,\mathrm{int}\,}}\left( C^1_0(\overline{\Omega })_+\right) ; \end{aligned}$$

• (b) left continuous.

Proof

(a) Let $0<\mu <\lambda \le \lambda ^*$. According to Proposition 3.2 we can find $u_\mu \in \mathcal {S}_\mu \subseteq {{\,\mathrm{int}\,}}\left( C^1_0(\overline{\Omega })_+\right)$ such that $u^*_\lambda -u_\mu \in {{\,\mathrm{int}\,}}\left( C^1_0(\overline{\Omega })_+\right)$. Since $u^*_\lambda \le u_\mu$ we obtain the desired conclusion.

(b) Suppose that $\lambda _n \rightarrow \lambda ^- \le \lambda ^*$. Then $\{u_n^*\}_{n\ge 1}:=\{u^*_{\lambda _n}\}_{n\ge 1}\subseteq {{\,\mathrm{int}\,}}\left( C^1_0(\overline{\Omega })_+\right)$ is increasing and

$$\begin{aligned} \underline{u} \le u_n^* \le u^*_{\lambda ^*}\quad \text {for all }n\in \mathbb {N}. \end{aligned}$$

(4.2)

From (4.2) and the nonlinear regularity theory of Lieberman [10] we have that $\{u^*_n\}_{n\ge 1}\subseteq C^1_0(\overline{\Omega })$ is relatively compact and so

$$\begin{aligned} u_n^*\rightarrow \tilde{u}^*_\lambda \quad \text {in }C^1_0(\overline{\Omega }). \end{aligned}$$

(4.3)

If $\tilde{u}^*_\lambda \ne u^*_\lambda$, then we can find $z_0\in \Omega$ such that

$$\begin{aligned} u^*_\lambda (z_0)<\tilde{u}^*_\lambda (z_0). \end{aligned}$$

From (4.3) we then derive

$$\begin{aligned} u^*_\lambda (z_0)<u_n^*(z_0)\quad \text {for all }n \ge n_0, \end{aligned}$$

which contradicts (a). So, $\tilde{u}^*_\lambda =u^*_\lambda$ and we conclude the left continuity of $\lambda \rightarrow u^*_\lambda$. $\square$

Summarizing our findings in this section, we can state the following theorem.

Theorem 4.3

If hypotheses H hold and $\lambda \in \mathcal {L}=(0,\lambda ^*]$, then problem (P$_\lambda$) admits a smallest positive solution $u^*_\lambda \in \mathcal {S}_\lambda \subseteq {{\,\mathrm{int}\,}}\left( C^1_0(\overline{\Omega })_+\right)$ and the map $\lambda \rightarrow u^*_\lambda$ from $\mathcal {L}=(0,\lambda ^*]$ into $C^1_0(\overline{\Omega })$ is

• (a) strictly increasing;

• (b) left continuous.