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

figure a

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.

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 \(\overline{u}\in {{\,\mathrm{int}\,}}\left( C^1_0(\overline{\Omega })_+\right) \). \(\square \)

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.

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.