## 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 (pq)-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 (pq)-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 (pq)-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 (pq)-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:

1. 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

1. (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}
2. (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}
3. (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}
4. (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}
5. (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 (pq)-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

1. (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}
2. (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}
3. (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

1. (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}
2. (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

1. (a)

strictly increasing;

2. (b)

left continuous.