(pq)-Equations with Singular and Concave Convex Nonlinearities

Abstract

We consider a nonlinear Dirichlet problem driven by the (pq)-Laplacian with \(1<q<p\). The reaction is parametric and exhibits the competing effects of a singular term and of concave and convex nonlinearities. We are looking for positive solutions and prove a bifurcation-type theorem describing in a precise way the set of positive solutions as the parameter varies. Moreover, we show the existence of a minimal positive solution and we study it as a function of the parameter.

Introduction

Let \(\Omega \subseteq \mathbb {R}^N\) be a bounded domain with a \(C^2\)-boundary \(\partial \Omega \). In this paper, we study the following parametric Dirichlet (pq)-equation

figurea

For \(r\in (1,\infty )\) we denote by \(\Delta _r\) the r-Laplace differential operator defined by

$$\begin{aligned} \Delta _r u ={{\,\mathrm{div}\,}}\left( |\nabla u|^{r-2} \nabla u\right) \quad \text {for all }u \in {W^{1,r}_0(\Omega )}. \end{aligned}$$

The perturbation in problem (P\(_\lambda \)), namely \(f:\Omega \times \mathbb {R}\rightarrow \mathbb {R}\), is a Carathéodory function, that is, f is measurable in the first argument and continuous in the second one. We suppose that \(f(x,\cdot )\) is \((p-1)\)-superlinear near \(+\infty \) but it does not satisfy the well-known Ambrosetti-Rabinowitz condition which we will write AR-condition for short. Hence, we have in problem (P\(_\lambda \)) the combined effects of singular terms (the function \(s\rightarrow \lambda s^{-\eta }\)), of sublinear (concave) terms (the function \(s\rightarrow \lambda s^{\tau -1}\) since \(1<\tau<q<p\)) and of superlinear (convex) terms (the function \(s\rightarrow f(x,s)\)). For the precise conditions on f we refer to hypotheses H(f) in Sect. 2. Consider the following two functions (for the sake of simplicity we drop the x-dependence)

$$\begin{aligned} f_1(s)=\left( s^+\right) ^{r-1}, \quad p<r<p^*, \qquad f_2(s) = {\left\{ \begin{array}{ll} \left( s^+\right) ^l&{}\text {if }s\le 1,\\ s^{p-1} \ln (s)+1&{}\text {if }1<s, \end{array}\right. } \quad q<l. \end{aligned}$$

Both functions satisfy our hypotheses H(f) but only \(f_1\) satisfies the AR-condition.

We are looking for positive solutions and we establish the precise dependence of the set of positive solutions of (P\(_\lambda \)) on the parameter \(\lambda >0\) as the latter varies. For the weight \(a(\cdot )\) we suppose the following assumptions

  1. H(a):

    \(a\in L^{\infty }(\Omega )\), \(a(x)\ge a_0>0\) for a.a. \(x\in \Omega \);

The main result in this paper is the following one.

Theorem 1.1

If hypotheses H(a) and H(f) hold, then there exists \(\lambda ^*\in (0,+\infty )\) such that

  1. (a)

    for all \(\lambda \in \left( 0,\lambda ^*\right) \), problem (P\(_\lambda \)) has at least two positive solutions

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

    for \(\lambda =\lambda ^*\), problem (P\(_\lambda \)) has at least one positive solution \(u^*\in {{\,\mathrm{int}\,}}\left( C^1_0(\overline{\Omega })_+\right) \);

  3. (c)

    for \(\lambda >\lambda ^*\), problem (P\(_\lambda \)) has no positive solution;

  4. (d)

    for every \(\lambda \in {\mathcal {L}}=\left( 0,\lambda ^*\right] \), problem (P\(_\lambda \)) has a smallest positive solution \(u^*_\lambda \in {{\,\mathrm{int}\,}}\left( C^1_0(\overline{\Omega })_+\right) \) and the map \(\lambda \rightarrow u^*_\lambda \) from \({\mathcal {L}}\) into \(C^1_0(\overline{\Omega })\) is strictly increasing, that is, \(0<\mu <\lambda \le \lambda ^*\) implies \(u^*_\lambda -u^*_\mu \in {{\,\mathrm{int}\,}}\left( C^1_0(\overline{\Omega })_+\right) \) and it is left continuous.

The study of elliptic problems with combined nonlinearities was initiated with the seminal paper of Ambrosetti–Brezis–Cerami [1] who studied semilinear Dirichlet equations driven by the Laplacian without any singular term. Their work has been extended to nonlinear problems driven by the p-Laplacian by García Azorero–Peral Alonso–Manfredi [5] and Guo–Zhang [11]. In both works there is no singular term and the reaction has the special form

$$\begin{aligned} x\rightarrow \lambda s^{\tau -1}+s^{r-1} \quad \text {for all}\, s \ge 0\, \text {with }1<\tau<p<r<p^*, \end{aligned}$$

where \(p^*\) is the critical Sobolev exponent to p given by

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

More recently there have been generalizations involving more general nonlinear differential operators, more general concave and convex nonlinearities and different boundary conditions. We refer to the works of Papageorgiou–Rădulescu–Repovš [23] for Robin problems and Papageorgiou–Winkert [19], Leonardi–Papageorgiou [14] and Marano–Marino–Papageorgiou [16] for Dirichlet problems. None of these works involves a singular term. Singular equations driven by the p-Laplacian and with a superlinear perturbation were investigated by Papageorgiou–Winkert [21].

We mention that (pq)-equations arise in many mathematical models of physical processes. We refer to Benci–D’Avenia–Fortunato–Pisani [2] for quantum physics and Cherfils-Il\('\)yasov [3] for reaction diffusion systems.

Finally, we mention recent papers which are very close to our topic dealing with certain types of nonhomogeneous and/or singular problems. We refer to Papageorgiou–Rădulescu–Repovš [26, 28], Papageorgiou–Zhang [22] and Ragusa–Tachikawa [30].

Preliminaries and Hypotheses

We denote by \(L^{p}(\Omega )\)\(\left( \text {or } L^p\left( \Omega ; \mathbb {R}^N\right) \right) \) and \(W^{1,p}_0(\Omega )\) the usual Lebesgue and Sobolev spaces with their norms \(\Vert \cdot \Vert _{p}\) and \(\Vert \cdot \Vert \), respectively. By means of the Poincaré inequality we have

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

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 known that

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

Furthermore, we need the ordered Banach space

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

and its 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

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

where \(n(\cdot )\) stands for the outward unit normal on \(\partial \Omega \). We will also use two more open cones. The first one is an open cone in the space \(C^1(\overline{\Omega })\) and is defined by

$$\begin{aligned} D_+&=\left\{ u \in C^1(\overline{\Omega })_+: u(x)>0 \text { for all } x\in \Omega , \ \frac{\partial u}{\partial n}\bigg |_{\partial \Omega \cap u^{-1}(0)}<0 \right\} . \end{aligned}$$

The second open cone is the interior of the order cone

$$\begin{aligned} K_+=\left\{ u\in C_0(\overline{\Omega }): u(x) \ge 0 \text { for all }x\in \overline{\Omega }\right\} \end{aligned}$$

of the Banach space

$$\begin{aligned} C_0(\overline{\Omega })=\left\{ u\in C(\overline{\Omega }) : u\big |_{\partial \Omega }=0\right\} . \end{aligned}$$

We know that

$$\begin{aligned} {{\,\mathrm{int}\,}}K_+=\left\{ u \in K_+: c_u \hat{d} \le u \text { for some }c_u>0\right\} \end{aligned}$$

with \({\hat{d}}(\cdot )=d(\cdot ,\partial \Omega )\). Let \({\hat{u}}_1\) denote the positive \(L^p\)-normalized, that is, \(\Vert {\hat{u}}_1\Vert _p=1\), eigenfunction of \(\left( -\Delta _p,W^{1,p}_0(\Omega )\right) \). We know that \(\hat{u}_1\in {{\,\mathrm{int}\,}}\left( C^1_0(\overline{\Omega })_+\right) \). From Papageorgiou–Rădulescu–Repovš [25] we have

$$\begin{aligned} c_u \hat{d}\le u \text { for some }c_u>0 \quad \text { if and only if }\quad \hat{c}_u\hat{u}_1 \le u \text { for some }\hat{c}_u>0. \end{aligned}$$

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

$$\begin{aligned}{}[u,v]&=\left\{ y\in W^{1,p}_0(\Omega ): u(x) \le y(x) \le v(x) \text { for a.\,a.\,}x\in \Omega \right\} ,\\ {{\,\mathrm{int}\,}}_{\mathop {_{C^1_0(\overline{\Omega })}}} [u,v]&=\text {the interior in } \mathop {C^1_0(\overline{\Omega })} \text { of } [u,v]\cap C^1_0(\overline{\Omega }),\\ [u)&= \left\{ y\in W^{1,p}_0(\Omega ): u(x) \le y(x) \text { for a.a. }x\in \Omega \right\} . \end{aligned}$$

If \(h,g \in L^{\infty }(\Omega )\), then we write \(h \prec g\) if and only if for every compact set \(K\subseteq \Omega \), there exists \(c_K>0\) such that \(c_K \le g(x)-h(x)\) for a.a. \(x\in K\). Note that if \(h,g \in C(\Omega )\) and \(h(x)<g(x)\) for all \(x\in \Omega \), then \(h\prec g\).

If X is a Banach space and \(\varphi \in C^1(X)\), then we denote by \(K_\varphi \) the critical set of \(\varphi \), that is,

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

Moreover, 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

$$\begin{aligned} \left( 1+\Vert u_n\Vert _X\right) \varphi '(u_n) \rightarrow 0\quad \text {in }X^* \text { as }n\rightarrow \infty , \end{aligned}$$

admits a strongly convergent subsequence.

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 defined by

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

This operator has the following properties, see Gasiński–Papageorgiou [8, p. 279].

Proposition 2.1

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

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

imply

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

The hypotheses on the function \(f(\cdot )\) are the following ones:

  1. H(f):

    \(f:\Omega \times \mathbb {R}\rightarrow \mathbb {R}\) is a Carathéodory function such that

    1. (i)
      $$\begin{aligned} 0\le f(x,s) \le c_1 \left[ 1+s^{r-1}\right] \end{aligned}$$

      for a. a. \(x\in \Omega \), for all  \(s \ge 0\)  with \(c_1>0\) and \(r\in (p,p^*)\);

    2. (ii)

      if \(F(x,s)=\int _0^sf(x,t)\,dt\), 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 \(\mu \in \left( (r-p)\max \left\{ 1,\frac{N}{p}\right\} ,p^*\right) \) with \(\mu >\tau \) such that

      $$\begin{aligned} 0<c_2 \le \liminf _{s\rightarrow +\infty } \frac{f(x,s)s-pF(x,s)}{s^\mu } \quad \text {uniformly for a.a.}\ x\in \Omega ; \end{aligned}$$
    4. (iv)
      $$\begin{aligned} \lim _{s\rightarrow 0^+} \frac{f(x,s)}{s^{q-1}}=0\quad \text {uniformly for a.a. }x\in \Omega ; \end{aligned}$$
    5. (v)

      for every \(\rho >0\) there exists \(\hat{\xi }_\rho >0\) such that the function

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

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

Remark 2.2

Since our aim is to produce positive solutions and all the hypotheses above concern the positive semiaxis \(\mathbb {R}_+=[0,+\infty )\), we may assume, without any loss of generality, that

$$\begin{aligned} f(x,s)=0 \quad \text {for a.a. }x\in \Omega \text { and for all }s \le 0. \end{aligned}$$
(2.1)

Note that hypothesis H(f)(iv) implies that \(f(x,0)=0\) for a.a. \(x\in \Omega \). From hypotheses H(f)(ii), (iii) we infer that

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

Therefore, the perturbation \(f(x,\cdot )\) is \((p-1)\)-superlinear for a.a. \(x\in \Omega \). However, the superlinearity of \(f(x,\cdot )\) is not expressed using the AR-condition which is common in the literature for superlinear problems. We recall that the AR-condition says that there exist \(\beta >p\) and \(M>0\) such that

$$\begin{aligned} 0&<\beta F(x,s) \le f(x,s)s \quad \text {for a.a. }x\in \Omega \text { and for all } s\ge M, \end{aligned}$$
(2.2)
$$\begin{aligned} 0&<\text {ess inf}_{x\in \Omega }\, F(x, M). \end{aligned}$$
(2.3)

In fact this is a uniliteral version of the AR-condition due to (2.1). Integrating (2.2) and using (2.3) gives the weaker condition

$$\begin{aligned} c_3 s^{\beta } \le F(x,s) \quad \text {for a.\,a.\,}x\in \Omega , \text { for all }x\ge M\text { and for some }c_3>0, \end{aligned}$$

which implies

$$\begin{aligned} c_3 s^{\beta -1} \le f(x,s) \quad \text {for a.\,a.\,}x\in \Omega \text { and for all }s \ge M. \end{aligned}$$

Hence, the AR-condition dictates that \(f(x,\cdot )\) eventually has at least \((\beta -1)\)-polynomial growth. In the present work we replace the AR-condition by hypothesis H(f)(iii) which includes in our framework also superlinear nonlinearities with slower growth near \(+\infty \).

Hypothesis H(f)(v) is a one-sided Hölder condition. If \(f(x,\cdot )\) is differentiable for a.a. \(x\in \Omega \) and if for every \(\rho >0\) there exists \(c_\rho >0\) such that

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

then hypothesis H(f)(v) is satisfied. We introduce the following sets

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

Moreover, we consider the following auxiliary Dirichlet problem

figureb

Proposition 2.3

If hypothesis H(a) holds, then for every \(\lambda >0\) problem (Q\(_\lambda \)) admits a unique solution \(\tilde{u}_\lambda \in {{\,\mathrm{int}\,}}\left( C^1_0(\overline{\Omega })_+\right) \).

Proof

We consider the \(C^1\)-functional \(\gamma _\lambda :W^{1,p}_0(\Omega )\rightarrow \mathbb {R}\) defined by

$$\begin{aligned} \gamma _\lambda (u)=\frac{1}{p}\Vert \nabla u\Vert _p^p +\frac{1}{q} \Vert \nabla u\Vert _q^q-\lambda \int _{\Omega }a(x) \left( u^+\right) ^{\tau }\,dx \quad \text {for all }u \in W^{1,p}_0(\Omega ). \end{aligned}$$

Since \(\tau<q<p\) it is clear that \(\gamma _\lambda :W^{1,p}_0(\Omega )\rightarrow \mathbb {R}\) is coercive and by the Sobolev embedding theorem, we see that \(\gamma _\lambda :W^{1,p}_0(\Omega )\rightarrow \mathbb {R}\) is sequentially weakly lower semicontinuous. Hence, there exists \({\tilde{u}}_\lambda \in W^{1,p}_0(\Omega )\) such that

$$\begin{aligned} \gamma _\lambda \left( \tilde{u}_\lambda \right) = \min \left[ \gamma _\lambda (u): u\in W^{1,p}_0(\Omega )\right] . \end{aligned}$$
(2.4)

If \(u \in {{\,\mathrm{int}\,}}\left( C^1_0(\overline{\Omega })_+\right) \) and \(t>0\) then

$$\begin{aligned} \gamma _\lambda (tu) = \frac{t^p}{p} \Vert \nabla u\Vert _p^p +\frac{t^q}{q} \Vert \nabla u\Vert ^q_q-\frac{\lambda t^\tau }{\tau } \int _{\Omega }a(x) u^2\,dx. \end{aligned}$$

Since \(\tau<q<p\), choosing \(t\in (0,1)\) small enough, we have \(\gamma _\lambda (tu)<0\) and so,

$$\begin{aligned} \gamma _\lambda \left( \tilde{u}_\lambda \right) <0=\gamma _\lambda (0), \end{aligned}$$

see (2.4), which shows that \({\tilde{u}}_\lambda \ne 0\). From (2.4) we know that \(\gamma _\lambda '\left( \tilde{u}_\lambda \right) =0\), that is,

$$\begin{aligned} \langle A_p \left( {\tilde{u}}_\lambda \right) ,h\rangle +\langle A_q\left( {\tilde{u}}_\lambda \right) ,h\rangle =\lambda \int _{\Omega }a(x) \left( {\tilde{u}}_\lambda ^+\right) ^{\tau -1}h\,dx \text { for all }h\in W^{1,p}_0(\Omega ). \end{aligned}$$
(2.5)

Choosing \(h=-\tilde{u}^-_\lambda \in W^{1,p}_0(\Omega )\) in (2.5) gives

$$\begin{aligned} \left\| \nabla {\tilde{u}}^-_\lambda \right\| _p^p +\left\| \nabla {\tilde{u}}_\lambda ^-\right\| _q^q =0, \end{aligned}$$

which shows that \({\tilde{u}}_\lambda \ge 0\) with \({\tilde{u}}_\lambda \ne 0\). Therefore, (2.5) becomes

$$\begin{aligned}&-\Delta _p {\tilde{u}}_\lambda -\Delta _q {\tilde{u}}_\lambda = \lambda a(x) {\tilde{u}}_\lambda ^{\tau -1}\quad \text {in } \Omega , \qquad {\tilde{u}}_\lambda \big |_{\partial \Omega }=0. \end{aligned}$$

We know that \(\tilde{u}_\lambda \in L^{\infty }(\Omega )\), see, for example Marino–Winkert [17]. Then, from the nonlinear regularity theory of Lieberman [15] we have that \({\tilde{u}}_\lambda \in C^1_0(\overline{\Omega })_+\setminus \{0\}\). Moreover, the nonlinear maximum principle of Pucci-Serrin [29, pp. 111, 120] implies that \({\tilde{u}}_\lambda \in {{\,\mathrm{int}\,}}\left( C^1_0(\overline{\Omega })_+\right) \).

We still have to show that this positive solution is unique. Suppose that \(\tilde{v}_\lambda \in W^{1,p}_0(\Omega )\) is another solution of (Q\(_\lambda \)). As before we can show that \(\tilde{v}_\lambda \in {{\,\mathrm{int}\,}}\left( C^1_0(\overline{\Omega })_+\right) \). We consider the integral functional \(j:L^{1}(\Omega )\rightarrow \overline{\mathbb {R}}=\mathbb {R}\cup \{+\infty \}\) defined by

$$\begin{aligned} j(u)= {\left\{ \begin{array}{ll} \frac{1}{p}\left\| \nabla u^{\frac{1}{q}}\right\| _p^p +\frac{1}{q} \left\| \nabla u^{\frac{1}{q}}\right\| ^q_q &{}\text {if } u\ge 0, u^{\frac{1}{q}} \in W^{1,p}_0(\Omega ),\\ +\infty &{}\text {otherwise}. \end{array}\right. } \end{aligned}$$

From Díaz–Saá [4, Lemma 1] we see that j is convex. Furthermore, applying Proposition 4.1.22 of Papageorgiou–Rădulescu–Repovš [24, p. 274], we obtain that

$$\begin{aligned} \frac{{\tilde{u}}_\lambda }{{\tilde{v}}_\lambda }, \frac{{\tilde{v}}_\lambda }{{\tilde{u}}_\lambda }\in L^{\infty }(\Omega ). \end{aligned}$$

We denote by

$$\begin{aligned} {{\,\mathrm{dom}\,}}j =\left\{ u\in L^{1}(\Omega ): j(u)<+\infty \right\} \end{aligned}$$

the effective domain of j and set \(h={\tilde{u}}_\lambda ^q-{\tilde{v}}_\lambda ^q\). One gets

$$\begin{aligned} {\tilde{u}}_\lambda ^q-th \in {{\,\mathrm{dom}\,}}j \quad \text {and}\quad {\tilde{v}}_\lambda ^q+th\in {{\,\mathrm{dom}\,}}j\quad \text {for all }t\in [0,1]. \end{aligned}$$

Note that the functional \(j:L^{1}(\Omega )\rightarrow \overline{\mathbb {R}}\) is Gateaux differentiable at \({\tilde{u}}_\lambda ^q\) and at \({\tilde{v}}_\lambda ^q\) in the direction h. Using the nonlinear Green’s identity, see Papageorgiou–Rădulescu–Repovš [24, Corollary 1.5.16, p. 34], we obtain

$$\begin{aligned} j'\left( {\tilde{u}}_\lambda ^q\right) (h)&=\frac{1}{q} \int _{\Omega }\frac{-\Delta _p {\tilde{u}}_\lambda -\Delta _q {\tilde{u}}_\lambda }{{\tilde{u}}_\lambda ^{q-1}}h\,dx =\frac{\lambda }{q}\int _{\Omega }\frac{a(x)}{{\tilde{u}}_\lambda ^{q-\tau }}h\,dx,\\ j'\left( {\tilde{v}}_\lambda ^q\right) (h)&=\frac{1}{q} \int _{\Omega }\frac{-\Delta _p {\tilde{v}}_\lambda -\Delta _q {\tilde{v}}_\lambda }{{\tilde{v}}_\lambda ^{q-1}}h\,dx =\frac{\lambda }{q}\int _{\Omega }\frac{a(x)}{{\tilde{v}}_\lambda ^{q-\tau }}h\,dx. \end{aligned}$$

The convexity of \(j:L^{1}(\Omega )\rightarrow \overline{\mathbb {R}}\) implies the monotonicity of \(j'\). Hence

$$\begin{aligned} 0 \le \frac{\lambda }{q}\int _{\Omega }a(x) \left[ \frac{1}{{\tilde{u}}_\lambda ^{q-\tau }}-\frac{1}{{\tilde{v}}_\lambda ^{q-\tau }} \right] \left[ {\tilde{u}}_\lambda ^q-{\tilde{v}}_\lambda ^q\right] \,dx\le 0, \end{aligned}$$

which implies \({\tilde{u}}_\lambda ={\tilde{v}}_\lambda \). Therefore, \({\tilde{u}}_\lambda \in {{\,\mathrm{int}\,}}\left( C^1_0(\overline{\Omega })_+\right) \) is the unique positive solution of the auxiliary problem (Q\(_\lambda \)). \(\square \)

This solution will provide a useful lower bound for the elements of the set of positive solutions \({\mathcal {S}}_\lambda \).

Positive Solutions

Let \(\tilde{u}_\lambda \in {{\,\mathrm{int}\,}}\left( C^1_0(\overline{\Omega })_+\right) \) be the unique positive solution of (Q\(_\lambda \)), see Proposition 2.3. Let \(s>N\). Then \(\tilde{u}_\lambda ^s\in {{\,\mathrm{int}\,}}K_+\) and so there exists \(c_4>0\) such that

$$\begin{aligned} {\hat{u}}_1 \le c_4 {\tilde{u}}_\lambda ^s, \end{aligned}$$

see Sect. 2. Hence

$$\begin{aligned} {\tilde{u}}_\lambda ^{-\eta } \le c_5 {\hat{u}}_1^{-\frac{\eta }{s}}\quad \text {for some }c_5>0. \end{aligned}$$

Applying the Lemma of Lazer–McKenna [13] we have

$$\begin{aligned} {\hat{u}}_1^{-\frac{\eta }{s}} \in L^{s}(\Omega ) \end{aligned}$$

and thus

$$\begin{aligned} {\tilde{u}}_\lambda ^{-\eta } \in L^{s}(\Omega ). \end{aligned}$$
(3.1)

We introduce the following modification of problem (P\(_\lambda \)) in which we have neutralized the singular term

figurec

Let \(\psi _\lambda :W^{1,p}_0(\Omega )\rightarrow \mathbb {R}\) be the Euler energy functional of problem (P\(_\lambda \)’) defined by

$$\begin{aligned} \psi _\lambda (u)&=\frac{1}{p}\Vert \nabla u\Vert _p^p +\frac{1}{q} \Vert \nabla u\Vert _q^q -\lambda \int _{\Omega }{\tilde{u}}_\lambda ^{-\eta } u\,dx\\&\quad -\frac{\lambda }{\tau } \int _{\Omega }a(x)\left( u^+\right) ^{\tau }\,dx-\int _{\Omega }F(x,u^+)\,dx \end{aligned}$$

for all \(u \in W^{1,p}_0(\Omega )\), see (3.1). It is clear that \(\psi _\lambda \in C^1(W^{1,p}_0(\Omega ))\).

Proposition 3.1

If hypotheses H(a) and H(f) hold and if \(\lambda >0\), then \(\psi _\lambda \) satisfies the C-condition.

Proof

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

$$\begin{aligned}&\left| \psi _\lambda (u_n)\right| \le c_6 \quad \text {for all }n\in \mathbb {N}\text { and for some }c_6>0,\end{aligned}$$
(3.2)
$$\begin{aligned}&(1+\Vert u_n\Vert )\psi '_\lambda (u_n)\rightarrow 0 \quad \text {in }W^{1,p}_0(\Omega )^*=W^{-1,p'}(\Omega ) \text { with }\frac{1}{p}+\frac{1}{p'}=1. \end{aligned}$$
(3.3)

From (3.3) we have

$$\begin{aligned}&\left| \langle A_p(u_n),h\rangle +\langle A_q(u_n),h\rangle -\lambda \int _{\Omega }\tilde{u}_\lambda ^{-\eta }h\,dx-\lambda \int _{\Omega }a(x) \left( u_n^+\right) ^{\tau -1}h\,dx\right. \nonumber \\&\quad \left. -\int _{\Omega }f\left( x,u_n^+\right) h\,dx\right| \le \frac{\varepsilon _n\Vert h\Vert }{1+\Vert u_n\Vert }\quad \text {for all } h\in W^{1,p}_0(\Omega ) \text { with }\varepsilon _n\rightarrow 0^+. \end{aligned}$$
(3.4)

Choosing \(h=-u_n^-\in W^{1,p}_0(\Omega )\) in (3.4) leads to

$$\begin{aligned} \left\| \nabla u_n^-\right\| _p^p \le \varepsilon _n\quad \text {for all }n \in \mathbb {N}, \end{aligned}$$

which implies

$$\begin{aligned} u_n^- \rightarrow 0\quad \text {in }W^{1,p}_0(\Omega ) \text { as }n\rightarrow \infty . \end{aligned}$$
(3.5)

Combining (3.2) and (3.5) gives

$$\begin{aligned}&\left\| \nabla u_n^+\right\| _p^p+\frac{p}{q}\left\| \nabla u_n^+\right\| _q^q -\lambda p\int _{\Omega }\tilde{u}_\lambda ^{-\eta } u_n^+\,dx-\frac{\lambda p}{\tau } \int _{\Omega }a(x) \left( u_n^+\right) ^{\tau }\,dx\nonumber \\&-\int _{\Omega }pF\left( x,u_n^+\right) \,dx \le c_7 \quad \text {for all }n\in \mathbb {N}\text { and for some }c_7>0. \end{aligned}$$
(3.6)

On the other hand, if we choose \(h=u_n^+\in W^{1,p}_0(\Omega )\) in (3.4), we obtain

$$\begin{aligned}&-\left\| \nabla u_n^+\right\| _p^p-\left\| \nabla u_n^+\right\| _q^q +\lambda \int _{\Omega }\tilde{u}_\lambda ^{-\eta } u_n^+\,dx+\lambda \int _{\Omega }a(x) \left( u_n^+\right) ^\tau \,dx\nonumber \\&\quad +\,\int _{\Omega }f\left( x,u_n^+\right) u_n^+\,dx \le \varepsilon _n \quad \text {for all }n\in \mathbb {N}. \end{aligned}$$
(3.7)

Adding (3.6) and (3.7) yields

$$\begin{aligned}&\int _{\Omega }\left[ f\left( x,u_n^+\right) u_n^+-pF\left( x,u_n^+\right) \right] \,dx\nonumber \\&\quad \le \,\lambda (p-1)\int _{\Omega }\tilde{u}_\lambda ^{-\eta }u_n^+\,dx+\lambda \left[ \frac{p}{\tau }-1\right] \int _{\Omega }a(x) \left( u_n^+\right) ^\tau \,dx. \end{aligned}$$
(3.8)

By hypotheses H(f)(i), (iii) we can find \(c_8>0\) such that

$$\begin{aligned} \frac{c_2}{2}s^\mu -c_8 \le f(x,s)s-pF(x,s)\quad \text {for a.a. } x\in \Omega \text { and for all }s\ge 0. \end{aligned}$$

This implies

$$\begin{aligned} \frac{c_2}{2}s^\mu \left\| u_n^+\right\| _\mu ^\mu -c_9 \le \int _{\Omega }\left[ f\left( x,u_n^+\right) u_n^+-pF\left( x,u_n^+\right) \right] \,dx \end{aligned}$$
(3.9)

for some \(c_9>0\) and for all \(n\in \mathbb {N}\).

Since \(s>N\) we have \(s'<N'\le p^*\). Hence, \(u_n^+\in L^{s'}(\Omega )\). Then, taking (3.1) along with Hölder’s inequality into account, we get

$$\begin{aligned} \lambda [p-1]\int _{\Omega }\tilde{u}_\lambda ^{-\eta }u_n^+\,dx \le c_{10} \left\| \tilde{u}_\lambda ^{-\eta }\right\| _s \left\| u_n^+\right\| _{s'} \end{aligned}$$
(3.10)

for some \(c_{10}=c_{10}(\lambda )>0\) and for all \(n\in \mathbb {N}\). Moreover, by hypothesis H(a), we have

$$\begin{aligned} \lambda \left[ \frac{p}{\tau }-1\right] \int _{\Omega }a(x) \left( u_n^+\right) ^\tau \,dx \le c_{11} \left\| u_n^+\right\| _\tau ^\tau \end{aligned}$$
(3.11)

for some \(c_{11}=c_{11}(\lambda )>0\) and for all \(n\in \mathbb {N}\).

Now we choose \(s>N\) large enough such that \(s'<\mu \). Returning to (3.8), using (3.9), (3.10) as well as (3.11) and using the fact that \(s', \tau <\mu \) by hypothesis H(f)(iii) leads to

$$\begin{aligned} \left\| u_n^+\right\| _\mu ^\mu \le c_{12}\left[ \left\| u_n^+\right\| _\mu +\left\| u_n^+\right\| _\mu ^\tau +1\right] \end{aligned}$$

for some \(c_{12}>0\) and for all \(n\in \mathbb {N}\). Since \(\tau <\mu \) we obtain

$$\begin{aligned} \left\{ u_n^+\right\} _{n \ge 1} \subseteq L^{\mu }(\Omega ) \text { is bounded}. \end{aligned}$$
(3.12)

Assume that \(N\ne p\). From hypothesis H(f)(iii) it is clear that we may assume \(\mu<r<p^*\). Then there exists \(t\in (0,1)\) such that

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

Taking the interpolation inequality into account, see Papageorgiou–Winkert [20, Proposition2.3.17, p. 116], we have

$$\begin{aligned} \left\| u_n^+\right\| _r\le \left\| u_n^+\right\| _\mu ^{1-t} \left\| u_n^+\right\| ^t_{p^*}, \end{aligned}$$

which by (3.12) implies that

$$\begin{aligned} \left\| u_n^+\right\| _r^r\le c_{13}\left\| u_n^+\right\| ^{tr} \end{aligned}$$
(3.13)

for some \(c_{13}>0\) and for all \(n\in \mathbb {N}\).

From hypothesis H(f)(i) we know that

$$\begin{aligned} f(x,s)s \le c_{14} \left[ 1+s^r\right] \end{aligned}$$
(3.14)

for a.a. \(x\in \Omega \), for all \(s\ge 0\) and for some \(c_{14}>0\). We choose \(h=u_n^+\in W^{1,p}_0(\Omega )\) in (3.4), that is,

$$\begin{aligned}&\left\| \nabla u_n^+\right\| _p^p+\left\| \nabla u_n^+\right\| _q^q -\lambda \int _{\Omega }\tilde{u}_\lambda ^{-\eta }u_n^+\,dx-\lambda \int _{\Omega }a(x) \left( u_n^+\right) ^{\tau }\,dx\\&\quad -\int _{\Omega }f\left( x,u_n^+\right) u_n^+\,dx\le \varepsilon _n\quad \text {for all } n\in \mathbb {N}. \end{aligned}$$

From this it follows by using (3.13), (3.14) and \(1<\tau<p<r\)

$$\begin{aligned} \left\| u_n^+\right\| ^p \le c_{15}\left[ 1+\left\| u_n^+\right\| ^{tr}\right] \end{aligned}$$
(3.15)

for some \(c_{15}>0\) and for all \(n\in \mathbb {N}\). The condition on \(\mu \), see hypothesis H(f)(iii), implies that \(tr<p\). Then from (3.15) we infer

$$\begin{aligned} \left\{ u_n^+\right\} _{n\ge 1} \subseteq W^{1,p}_0(\Omega ) \text { is bounded.} \end{aligned}$$
(3.16)

If \(N=p\), then we have by definition \(p^*=\infty \). The Sobolev embedding theorem ensures that \(W^{1,p}_0(\Omega )\hookrightarrow L^{\vartheta }(\Omega )\) for all \(1\le \vartheta <\infty \). So, in order to apply the previous arguments we need to replace \(p^*\) by \(\vartheta>r>\mu \) and choose \(t \in (0,1)\) such that

$$\begin{aligned} \frac{1}{r}=\frac{1-t}{\mu }+\frac{t}{\vartheta }, \end{aligned}$$

which implies

$$\begin{aligned} tr=\frac{\vartheta (r-\mu )}{\vartheta -\mu }. \end{aligned}$$

Note that \(\frac{\vartheta (r-\mu )}{\vartheta -\mu }\rightarrow r-\mu <p\) as \(\vartheta \rightarrow +\infty \). So, for \(\vartheta >r\) large enough, we see that \(tr<p\) and again (3.16) holds.

From (3.5) and (3.16) we infer that

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

So, we may assume that

$$\begin{aligned} u_n\overset{{{\,\mathrm{w}\,}}}{\rightarrow }u \quad \text {in }W^{1,p}_0(\Omega ) \quad \text {and}\quad u_n\rightarrow u \quad \text {in }L^{r}(\Omega ). \end{aligned}$$
(3.17)

We choose \(h=u_n-u\in W^{1,p}_0(\Omega )\) in (3.4), pass to the limit as \(n\rightarrow \infty \) and use the convergence properties in (3.17). This gives

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

and since \(A_q\) is monotone we obtain

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

By (3.16) we then conclude that

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

Applying Proposition 2.1 shows that \(u_n\rightarrow u\) in \(W^{1,p}_0(\Omega )\) and so we conclude that \(\psi _\lambda \) satisfies the C-condition. \(\square \)

Proposition 3.2

If hypotheses H(a) and H(f) hold, then there exists \(\hat{\lambda }>0\) such that for every \(\lambda \in \left( 0,\hat{\lambda }\right) \) we can find \(\rho _\lambda >0\) for which we have

$$\begin{aligned} \psi _\lambda (0)=0<\inf \left[ \psi _\lambda (u): \Vert u\Vert =\rho _\lambda \right] =m_\lambda . \end{aligned}$$

Proof

Hypotheses H(f)(i), (iv) imply that for a given \(\varepsilon >0\) we can find \(c_{16}=c_{16}(\varepsilon )>0\) such that

$$\begin{aligned} F(x,s)\le \frac{\varepsilon }{q}s^q +c_{16}s^r \quad \text {for a.a. }x\in \Omega \text { and for all }s\ge 0. \end{aligned}$$
(3.18)

Recall that \(\tilde{u}_\lambda ^{-\eta } \in L^{s}(\Omega )\) with \(s>N\), see (3.1). We choose \(s>N\) large enough such that \(s'<p^*\). Then, by Hölder’s inequality, we have

$$\begin{aligned} \lambda \int _{\Omega }{\tilde{u}}_\lambda ^{-\eta }u\,dx \le \lambda c_{17}\Vert u\Vert \quad \text {for some }c_{17}>0. \end{aligned}$$
(3.19)

Moreover, one gets

$$\begin{aligned} \frac{\lambda }{\tau } \int _{\Omega }a(x) |u|^\tau \,dx\le \frac{\lambda \Vert a\Vert _\infty }{\tau } \Vert u\Vert ^\tau . \end{aligned}$$
(3.20)

Applying (3.18), (3.19) and (3.20) leads to

$$\begin{aligned} \psi _\lambda (u) \ge \frac{1}{p} \Vert \nabla u\Vert _p^p +\frac{1}{q} \left[ \Vert \nabla u\Vert _q^q-\varepsilon \Vert u\Vert _q^q\right] -c_{18} \left[ \Vert u\Vert ^r+\lambda \left( \Vert u\Vert +\Vert u\Vert ^\tau \right) \right] \end{aligned}$$
(3.21)

for some \(c_{18}>0\). Let \({\hat{\lambda }}_1(q)>0\) be the principal eigenvalue of \(\left( -\Delta _q,W^{1,q}_0(\Omega )\right) \). Then, from the variational characterization of \(\hat{\lambda }_1(q)\), see Gasiński–Papageorgiou [6, p. 732], we obtain

$$\begin{aligned} \frac{1}{q} \left[ \Vert \nabla u\Vert _q^q-\varepsilon \Vert u\Vert _q^q\right] \ge \frac{1}{q} \left[ 1-\frac{\varepsilon }{\hat{\lambda }_1(q)}\right] \Vert \nabla u\Vert _q^q. \end{aligned}$$

Choosing \(\varepsilon \in \left( 0,\hat{\lambda }_1(q)\right) \) we infer that

$$\begin{aligned} \frac{1}{q} \left[ \Vert \nabla u\Vert _q^q-\varepsilon \Vert u\Vert _q^q\right] >0. \end{aligned}$$
(3.22)

Since \(1<\tau <r\), it holds

$$\begin{aligned} \Vert u\Vert ^\tau \le \Vert u\Vert +\Vert u\Vert ^r. \end{aligned}$$
(3.23)

Applying (3.22) and (3.23) to (3.21) gives

$$\begin{aligned} \psi _\lambda (u)&\ge \frac{1}{p} \Vert u\Vert ^p -c_{18} \left[ 2\lambda \Vert u\Vert +(\lambda +1)\Vert u\Vert ^{r}\right] \nonumber \\&\ge \left[ \frac{1}{p} -c_{18} \left( 2\lambda \Vert u\Vert ^{1-p}+(\lambda +1)\Vert u\Vert ^{r-p}\right) \right] \Vert u\Vert ^p. \end{aligned}$$
(3.24)

We consider now the function

$$\begin{aligned} k_\lambda (t)=2\lambda t^{1-p}+(\lambda +1)t^{r-p}\quad \text {for all }t>0. \end{aligned}$$

It is clear that \(k_\lambda \in C^1(0,\infty )\) and since \(1<p<r\) we see that

$$\begin{aligned} k_\lambda (t)\rightarrow +\infty \quad \text {as }t\rightarrow 0^+ \text { and as }t\rightarrow +\infty . \end{aligned}$$

Hence, there exists \(t_0>0\) such that

$$\begin{aligned} k_\lambda (t_0)=\min \left[ k_\lambda (t): t>0\right] , \end{aligned}$$

which implies that \(k_\lambda '(t_0)=0\). Therefore,

$$\begin{aligned} 2\lambda (p-1)t_0^{-p}=(r-p)(\lambda +1)t_0^{r-p-1}. \end{aligned}$$

From this we deduce that

$$\begin{aligned} t_0=t_0(\lambda )=\left[ \frac{2\lambda (p-1)}{(r-p)(\lambda +1)}\right] ^{\frac{1}{r-1}}. \end{aligned}$$

We have

$$\begin{aligned} k_\lambda (t_0) =2\lambda \frac{(r-p)(\lambda +1)^{\frac{p-1}{r-1}}}{(2\lambda (p-1))^{\frac{p-1}{r-1}}} +(\lambda +1) \frac{(2\lambda (p-1))^{\frac{r-p}{r-1}}}{((r-p)(\lambda +1))^{\frac{r-p}{r-1}}}. \end{aligned}$$

Since \(1<p<r\) we see that

$$\begin{aligned} k_\lambda (t_0)\rightarrow 0 \quad \text {as }\lambda \rightarrow 0^+. \end{aligned}$$

Therefore, we can find \(\hat{\lambda }>0\) such that

$$\begin{aligned} k_\lambda (t_0)<\frac{1}{pc_{18}}\quad \text {for all }\lambda \in \left( 0,\hat{\lambda }\right) . \end{aligned}$$

Then, by (3.24) we see that

$$\begin{aligned} \psi _\lambda (u)>0=\psi _\lambda (0)\quad \text {for all }\Vert u\Vert =t_0(\lambda )=\rho _\lambda \text { and for all }\lambda \in \left( 0,\hat{\lambda }\right) . \end{aligned}$$

From hypothesis H(f)(ii) we see that for every \(u\in {{\,\mathrm{int}\,}}\left( C^1_0(\overline{\Omega })_+\right) \) we have

$$\begin{aligned} \psi _\lambda (tu)\rightarrow -\infty \quad \text {as }t\rightarrow +\infty . \end{aligned}$$
(3.25)

Proposition 3.3

If hypotheses H(a) and H(f) hold and if \(\lambda \in \left( 0,\hat{\lambda }\right) \), then problem (P\(_\lambda \)’) admits a solution \(\overline{u}_\lambda \in {{\,\mathrm{int}\,}}\left( C^1_0(\overline{\Omega })_+\right) \).

Proof

Propositions 3.1, 3.2 and (3.25) permit the use of the mountain pass theorem. So, we can find \(\overline{u}_\lambda \in W^{1,p}_0(\Omega )\) such that

$$\begin{aligned} \overline{u}_\lambda \in K_{\psi _\lambda }\quad \text {and}\quad \psi _\lambda (0)=0<m_\lambda \le \psi _\lambda (\overline{u}_\lambda ). \end{aligned}$$
(3.26)

From (3.26) we see that \(\overline{u}_\lambda \ne 0\) and \(\psi _\lambda '(\overline{u}_\lambda )=0\), that is,

$$\begin{aligned}&\langle A_p(\overline{u}_\lambda ),h\rangle +\langle A_q(\overline{u}_\lambda ),h\rangle \nonumber \\&=\lambda \int _{\Omega }\tilde{u}_\lambda ^{-\eta }h\,dx+\lambda \int _{\Omega }a(x) \left( \overline{u}_\lambda ^+\right) ^{\tau -1}h\,dx +\int _{\Omega }f\left( x,\overline{u}_\lambda ^+\right) h\,dx \end{aligned}$$
(3.27)

for all \(h\in W^{1,p}_0(\Omega )\). We choose \(h=-\overline{u}_\lambda ^-\in W^{1,p}_0(\Omega )\) in (3.27) which shows that

$$\begin{aligned} \left\| \overline{u}_\lambda ^-\right\| ^p \le 0. \end{aligned}$$

Thus, \(\overline{u}_\lambda \ge 0\) with \(\overline{u}_\lambda \ne 0\).

From (3.27) we know that \(\overline{u}_\lambda \) is a positive solution of (P\(_\lambda \)’) with \(\lambda \in \left( 0,{\hat{\lambda }}\right) \). This means

$$\begin{aligned}&-\Delta _p \overline{u}_\lambda -\Delta _q \overline{u}_\lambda = \lambda \tilde{u}_\lambda ^{-\eta }+\lambda a(x)\overline{u}_\lambda ^{\tau -1}+f(x,\overline{u}_\lambda )\quad \text {in } \Omega , \quad \overline{u}_\lambda \big |_{\partial \Omega }=0. \end{aligned}$$

As before, see the proof of Proposition 2.3, using the nonlinear regularity theory, we have \(\overline{u}_\lambda \in C^1_0(\overline{\Omega })_+\setminus \{0\}\). The nonlinear maximum principle, see Pucci–Serrin [29, pp. 111, 120] implies that \(\overline{u}_\lambda \in {{\,\mathrm{int}\,}}\left( C^1_0(\overline{\Omega })_+\right) \).

Proposition 3.4

If hypotheses H(a) and H(f) hold and if \(\lambda \in \left( 0,{\hat{\lambda }}\right) \), then \({\tilde{u}}_\lambda \le \overline{u}_\lambda \).

Proof

We introduce the Carathéodory function \(g_\lambda :\Omega \times \mathbb {R}\rightarrow \mathbb {R}\) defined by

$$\begin{aligned} g_\lambda (x,s)= {\left\{ \begin{array}{ll} \lambda a(x) \left( s^+\right) ^{\tau -1} &{}\text {if }s\le \overline{u}_\lambda (x),\\ \lambda a(x) \overline{u}_\lambda (x)^{\tau -1} &{}\text {if }\overline{u}_\lambda (x)<s. \end{array}\right. } \end{aligned}$$
(3.28)

We set \(G_\lambda (x,s)=\int ^s_0g_\lambda (x,t)\,dt\) and consider the \(C^1\)-functional \(\sigma _\lambda :W^{1,p}_0(\Omega )\rightarrow \mathbb {R}\) defined by

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

From (3.28) it is clear that \(\sigma _\lambda :W^{1,p}_0(\Omega )\rightarrow \mathbb {R}\) is coercive. Moreover, by the Sobolev embedding, we have that \(\sigma _\lambda :W^{1,p}_0(\Omega )\rightarrow \mathbb {R}\) is sequentially weakly lower semicontinuous. Then, by the Weierstraß-Tonelli theorem, we can find \({\hat{u}}_\lambda \in W^{1,p}_0(\Omega )\) such that

$$\begin{aligned} \sigma _\lambda \left( {\hat{u}}_\lambda \right) = \min \left[ \sigma _\lambda (u):u\in W^{1,p}_0(\Omega )\right] . \end{aligned}$$
(3.29)

Since \(\tau<q<p\), we have \(\sigma _\lambda \left( {\hat{u}}_\lambda \right) <0=\sigma _\lambda (0)\) which implies \({\hat{u}}_\lambda \ne 0\).

From (3.29) we have \(\sigma _\lambda '\left( \hat{u}_\lambda \right) =0\), that is,

$$\begin{aligned} \langle A_p \left( {\hat{u}}_\lambda \right) ,h\rangle +\langle A_q\left( {\hat{u}}_\lambda \right) ,h\rangle =\int _{\Omega }g_\lambda \left( x,{\hat{u}}_\lambda \right) h\,dx\quad \text {for all }h\in W^{1,p}_0(\Omega ). \end{aligned}$$
(3.30)

First, we choose \(h=-{\hat{u}}_\lambda ^-\in W^{1,p}_0(\Omega )\) in (3.30). Then, by the definition of the truncation in (3.28) we easily see that \(\Vert {\hat{u}}_\lambda ^-\Vert ^p \le 0\) and so, \({\hat{u}}_\lambda \ge 0\) with \({\hat{u}}_\lambda \ne 0\).

Next, we choose \(h=\left( {\hat{u}}_\lambda -\overline{u}_\lambda \right) ^+\in W^{1,p}_0(\Omega )\) in (3.30) which gives, due to (3.28) and \(f\ge 0\),

$$\begin{aligned}&\langle A_p\left( {\hat{u}}_\lambda \right) ,\left( {\hat{u}}_\lambda -\overline{u}_\lambda \right) ^+\rangle +\langle A_q\left( {\hat{u}}_\lambda \right) ,\left( {\hat{u}}_\lambda -\overline{u}_\lambda \right) ^ +\rangle \\&\quad = \int _{\Omega }\lambda a(x) \overline{u}_\lambda ^{\tau -1} \left( {\hat{u}}_\lambda -\overline{u}_\lambda \right) ^+\,dx\\&\quad \le \int _{\Omega }\left[ \lambda {\tilde{u}}_\lambda ^{-\eta } +\lambda a(x) \overline{u}_\lambda ^{\tau -1}+f\left( x,\overline{u}_\lambda \right) \right] \left( {\hat{u}}_\lambda -\overline{u}_\lambda \right) ^+\,dx\\&\quad = \langle A_p\left( \overline{u}_\lambda \right) ,\left( {\hat{u}}_\lambda -\overline{u}_\lambda \right) ^+\rangle +\langle A_q\left( \overline{u}_\lambda \right) ,\left( {\hat{u}}_\lambda -\overline{u}_\lambda \right) ^+\rangle . \end{aligned}$$

This shows that \({\hat{u}}_\lambda \le \overline{u}_\lambda \). We have proved that

$$\begin{aligned} {\hat{u}}_\lambda \in \left[ 0,\overline{u}_\lambda \right] , {\hat{u}}_\lambda \ne 0. \end{aligned}$$

Hence, \({\hat{u}}_\lambda \) is a positive solution of (Q\(_\lambda \)) and due to Proposition 2.3 we know that \({\hat{u}}_\lambda ={\tilde{u}}_\lambda \in {{\,\mathrm{int}\,}}\left( C^1_0(\overline{\Omega })_+\right) \). Therefore, \({\tilde{u}}_\lambda \le \overline{u}_\lambda \) for all \(\lambda \in \left( 0,{\hat{\lambda }}\right) \). \(\square \)

Now we are able to establish the nonemptiness of the set \({\mathcal {L}}\) (being the set of all admissible parameters) determine the regularity of the elements in the solution set \({\mathcal {S}}_\lambda \).

Proposition 3.5

If hypotheses H(a) and H(f) hold, then \({\mathcal {L}}\ne \emptyset \) and, for every \(\lambda >0\), \({\mathcal {S}}_\lambda \subseteq {{\,\mathrm{int}\,}}\left( C^1_0(\overline{\Omega })_+\right) \).

Proof

Let \(\lambda \in \left( 0,{\hat{\lambda }}\right) \). From Proposition 3.4 we know that \({\tilde{u}}_\lambda \le \overline{u}_\lambda \). So we can define the truncation \(e_\lambda :\Omega \times \mathbb {R}\rightarrow \mathbb {R}\) of the reaction of problem (P\(_\lambda \))

$$\begin{aligned}&e_\lambda (x,s)\nonumber \\&= {\left\{ \begin{array}{ll} \lambda \left[ \tilde{u}_\lambda (x)^{-\eta }+a(x)\tilde{u}_\lambda (x)^{\tau -1}\right] +f\left( x,\tilde{u}_\lambda (x)\right) &{}\text {if }s<\tilde{u}_\lambda (x),\\ \lambda \left[ s^{-\eta }+a(x)s^{\tau -1}\right] +f(x,s) &{}\text {if }\tilde{u}_\lambda (x) \le s\le \overline{u}_\lambda (x),\\ \lambda \left[ \overline{u}_\lambda (x)^{-\eta }+a(x)\overline{u}_\lambda (x)^{\tau -1}\right] +f\left( x,\overline{u}_\lambda (x)\right) &{}\text {if }\overline{u}_\lambda (x) <s. \end{array}\right. } \end{aligned}$$
(3.31)

This is a Carathéodory function. We set \(E_\lambda (x,s)=\int ^s_0 e_{\lambda }(x,t)\,dt\) and consider the \(C^1\)-functional \(J_\lambda :W^{1,p}_0(\Omega )\rightarrow \mathbb {R}\) defined by

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

From (3.31) we see that \(J_\lambda :W^{1,p}_0(\Omega )\rightarrow \mathbb {R}\) is coercive and the Sobolev embedding theorem implies that J is also sequentially weakly lower semicontinuous. Hence, its global minimizer \(u_\lambda \in W^{1,p}_0(\Omega )\) exists, that is,

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

Hence, \(J_\lambda '(u_\lambda )=0\) which means that

$$\begin{aligned} \langle A_p\left( u_\lambda \right) ,h\rangle +\langle A_q\left( u_\lambda \right) ,h\rangle =\int _{\Omega }e_\lambda \left( x,u_\lambda \right) h\,dx \quad \text {for all } h \in W^{1,p}_0(\Omega ). \end{aligned}$$
(3.32)

We choose \(h=\left( u_\lambda -\overline{u}_\lambda \right) ^+\in W^{1,p}_0(\Omega )\) in (3.32). Then, by using (3.31) and Propositions 3.4 and 3.3 we obtain

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

This shows that \(u_\lambda \le \overline{u}_\lambda \).

Next, we choose \(h=\left( \tilde{u}_\lambda -u_\lambda \right) ^+\in W^{1,p}_0(\Omega )\) in (3.32). Then, by (3.31) and hypotheses H(a) as well as H(f)(i) it follows

$$\begin{aligned}&\langle A_p \left( u_\lambda \right) ,\left( \tilde{u}_\lambda -u_\lambda \right) ^+\rangle +\langle A_q\left( u_\lambda \right) ,\left( \tilde{u}_\lambda -u_\lambda \right) ^+\rangle \\&\quad =\, \int _{\Omega }\left( \lambda \left[ {\tilde{u}}^{-\eta }+a(x){\tilde{u}}_\lambda ^{\tau -1}\right] +f\left( x,{\tilde{u}}_\lambda \right) \right) \left( {\tilde{u}}_\lambda -u_\lambda \right) ^+\,dx\\&\qquad \ge \int _{\Omega }\lambda {\tilde{u}}_\lambda ^{-\eta } \left( {\tilde{u}}_\lambda -u_\lambda \right) ^+dx\\&\qquad = \langle A_p\left( {\tilde{u}}_\lambda \right) ,\left( {\tilde{u}}_\lambda -u_\lambda \right) ^+\rangle +\langle A_q\left( {\tilde{u}}_\lambda \right) ,\left( {\tilde{u}}_\lambda -u_\lambda \right) ^+\rangle . \end{aligned}$$

Hence, \({\tilde{u}}_\lambda \le u_\lambda \) and so we have proved that \(u_\lambda \in \left[ {\tilde{u}}_\lambda , \overline{u}_\lambda \right] \). Then, with view to (3.31) and (3.32), we see that \(u_\lambda \) is a positive solution of (P\(_\lambda \)) for \(\lambda \in \left( 0,{\hat{\lambda }}\right) \). In particular, we have

$$\begin{aligned} -\Delta _p u_\lambda (x)-\Delta _qu_\lambda (x)=\lambda u_\lambda (x)^{-\eta }+a_\lambda (x)u_\lambda (x)^{\tau -1}+f(x,u_\lambda (x))\quad \text {for a.a. }x\in \Omega . \end{aligned}$$

The nonlinear regularity theory, see Lieberman [15], and the nonlinear maximum principle, see Pucci–Serrin [29, pp. 111 and 120] imply that \(u_\lambda \in {{\,\mathrm{int}\,}}\left( C^1_0(\overline{\Omega })_+\right) \).

Concluding we can say that \(\left( 0,{\hat{\lambda }}\right) \subseteq {\mathcal {L}}\) which means that \({\mathcal {L}}\) is nonempty. Moreover, for all \(\lambda >0\), \({\mathcal {S}}_\lambda \subseteq {{\,\mathrm{int}\,}}\left( C^1_0(\overline{\Omega })_+\right) \). \(\square \)

Reasoning as in the proof of Proposition 3.4 with \(\overline{u}_\lambda \) replaced by \(u \in {\mathcal {S}}_\lambda \subseteq {{\,\mathrm{int}\,}}\left( C^1_0(\overline{\Omega })_+\right) \), we obtain the following result.

Proposition 3.6

If hypotheses H(a) and H(f) hold and if \(\lambda \in {\mathcal {L}}\), then \({\tilde{u}}_\lambda \le u\) for all \(u \in {\mathcal {S}}_\lambda \).

Moreover, the map \(\lambda \rightarrow {\tilde{u}}_\lambda \) from \((0,+\infty )\) into \(C^1_0(\overline{\Omega })\) exhibits a strong monotonicity property which we will use in the sequel.

Proposition 3.7

If hypotheses H(a) holds and if \(0<\lambda <\lambda '\), then \({\tilde{u}}_{\lambda '}-{\tilde{u}}_\lambda \in {{\,\mathrm{int}\,}}\left( C^1_0(\overline{\Omega })_+\right) \).

Proof

Following the proof of Proposition 3.4 we can show that

$$\begin{aligned} {\tilde{u}}_\lambda \le {\tilde{u}}_{\lambda '}. \end{aligned}$$
(3.33)

From (3.33) we have

$$\begin{aligned} -\Delta _p {\tilde{u}}_\lambda -\Delta _q{\tilde{u}}_\lambda&=\lambda a(x) {\tilde{u}}_\lambda ^{\tau -1}\nonumber \\&=\lambda 'a(x){\tilde{u}}_\lambda ^{\tau -1}-\left( \lambda '-\lambda \right) {\tilde{u}}_\lambda ^{\tau -1}\nonumber \\&\le \lambda 'a(x){\tilde{u}}_{\lambda '}^{\tau -1}\nonumber \\&= -\Delta _p {\tilde{u}}_{\lambda '}-\Delta _q{\tilde{u}}_{\lambda '}. \end{aligned}$$
(3.34)

Note that \(0\prec \left( \lambda '-\lambda \right) \tilde{u}_\lambda ^{\tau -1}\). So, from (3.34) and Gasiński–Papageorgiou [9, Proposition 3.2], we have

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

\(\square \)

Next we are going to show that \({\mathcal {L}}\) is an interval.

Proposition 3.8

If hypotheses H(a) and H(f) hold and if \(\lambda \in {\mathcal {L}}\) and \(\mu \in (0,\lambda )\), then \(\mu \in {\mathcal {L}}\).

Proof

Since \(\lambda \in {\mathcal {L}}\) there exists \(u_\lambda \in {\mathcal {S}}_\lambda \subseteq {{\,\mathrm{int}\,}}\left( C^1_0(\overline{\Omega })_+\right) \), see Proposition 3.5. From Propositions 3.4 and 3.7 we have

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

We introduce the truncation function \({\hat{k}}_\mu :\Omega \times \mathbb {R}\rightarrow \mathbb {R}\) defined by

$$\begin{aligned} {\hat{k}}_\mu (x,s) = {\left\{ \begin{array}{ll} \mu \left[ {\tilde{u}}_\mu (x)^{-\eta }+a(x)u_\mu (x)^{\tau -1}\right] +f\left( x,u_\mu (x)\right) &{}\text {if }s<{\tilde{u}}_\mu (x),\\ \mu \left[ s^{-\eta }+a(x)s^{\tau -1}\right] +f\left( x,s\right) &{}\text {if } {\tilde{u}}_\mu (x)\le s \le u_\lambda (x),\\ \mu \left[ u_\lambda (x)^{-\eta }+a(x)u_\lambda (x)^{\tau -1}\right] +f\left( x,u_\lambda (x)\right) &{}\text {if }u_\lambda (x)<s, \end{array}\right. } \end{aligned}$$
(3.35)

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

$$\begin{aligned} {\hat{\sigma }}_\mu (u)=\frac{1}{p}\Vert \nabla u\Vert ^p_p+\frac{1}{q}\Vert \nabla u\Vert _q^q-\int _{\Omega }{\hat{K}}_\mu (x,u)\,dx\quad \text {for all }u \in W^{1,p}_0(\Omega ). \end{aligned}$$

This functional is coercive because of (3.35) and sequentially weakly lower semicontinuous due to the Sobolev embedding theorem. Hence, there exists \(u_\mu \in W^{1,p}_0(\Omega )\) such that

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

Therefore, \({\hat{\sigma }}_\mu '(u_\mu )=0\) and so

$$\begin{aligned} \langle A_p\left( u_\mu \right) ,h\rangle +\langle A_q\left( u_\mu \right) ,h\rangle =\int _{\Omega }{\hat{k}}_\mu \left( x,u_\mu \right) h\,dx \end{aligned}$$
(3.36)

for all \(h\in W^{1,p}_0(\Omega )\). We first choose \(h=\left( u_\mu -u_\lambda \right) ^+\in W^{1,p}_0(\Omega )\) in (3.36). Then, by (3.35), \(\mu <\lambda \) and since \(u_\lambda \in {\mathcal {S}}_\lambda \), we obtain

$$\begin{aligned}&\langle A_p\left( u_\mu \right) ,\left( u_\mu -u_\lambda \right) ^+\rangle +\langle A_q\left( u_\mu \right) , \left( u_\mu -u_\lambda \right) ^+\rangle \\&\quad =\int _{\Omega }\left[ \mu \left( u_\mu ^{-\eta }+a(x)u_\lambda ^{\tau -1}\right) +f\left( x,u_\lambda \right) \right] \left( u_\mu -u_\lambda \right) ^+\,dx\\&\quad \le \int _{\Omega }\left[ \lambda \left( u_\lambda ^{-\eta }+a(x)u_\lambda ^{\tau -1}\right) +f\left( x,u_\lambda \right) \right] \left( u_\mu -u_\lambda \right) ^+\,dx\\&\quad =\langle A_p\left( u_\lambda \right) , \left( u_\mu -u_\lambda \right) ^+\rangle +\langle A_q\left( u_\lambda \right) ,\left( u_\mu -u_\lambda \right) ^+\rangle . \end{aligned}$$

Hence, \(u_\mu \le v_\lambda \). In the same way, choosing \(h=\left( {\tilde{u}}_\mu -u_\mu \right) ^+ \in W^{1,p}_0(\Omega )\), we get from (3.35), hypotheses H(a), H(f)(i) and Proposition 2.3 that

$$\begin{aligned}&\langle A_p\left( u_\mu \right) ,\left( {\tilde{u}}_\mu -u_\mu \right) ^+\rangle +\langle A_q\left( u_\mu \right) , \left( {\tilde{u}}_\mu -u_\mu \right) ^+\rangle \\&\quad =\int _{\Omega }\left[ \mu \left( {\tilde{u}}_\mu ^{-\eta }+a(x){\tilde{u}}_\mu ^{\tau -1}\right) +f\left( x,{\tilde{u}}_\mu \right) \right] \left( {\tilde{u}}_\mu -u_\mu \right) ^+ \,dx\\&\qquad \ge \int _{\Omega }\mu {\tilde{u}}_\mu ^{-\eta } \left( {\tilde{u}}_\mu -u_\mu \right) ^+\,dx\\&\qquad = \langle A_p\left( {\tilde{u}}_\mu \right) ,\left( {\tilde{u}}_\mu -u_\mu \right) ^+\rangle +\langle A_q\left( {\tilde{u}}_\mu \right) ,\left( {\tilde{u}}_\mu -u_\mu \right) ^+\rangle . \end{aligned}$$

Thus, \({\tilde{u}}_\mu \le u_\mu \). We have proved that

$$\begin{aligned} u_\mu \in \left[ \tilde{u}_\mu ,u_\lambda \right] . \end{aligned}$$
(3.37)

From (3.37), (3.35) and (3.36) it follows that

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

\(\square \)

Now we are going to prove that the solution multifunction \(\lambda \rightarrow {\mathcal {S}}_\lambda \) has a kind of weak monotonicity property.

Proposition 3.9

If hypotheses H(a) and H(f) hold and if \(\lambda \in {\mathcal {L}}, u_\lambda \in \mathcal {S}_\lambda \subseteq {{\,\mathrm{int}\,}}\left( C^1_0(\overline{\Omega })_+\right) \) 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 Proposition 3.8 and its proof we know that \(\mu \in {\mathcal {L}}\) and that we can find \(u_\mu \in {\mathcal {S}}_\mu \subseteq {{\,\mathrm{int}\,}}\left( C^1_0(\overline{\Omega })_+\right) \) such that \(u_\mu \le v_\lambda \). Let \(\rho =\Vert u_\lambda \Vert _\infty \) and let \(\hat{\xi }_\rho >0\) be as postulated by hypothesis H(f)(v). Using \(u_\mu \in {\mathcal {S}}_{\mu }\), hypotheses H(a), H(f)(v) and recalling that \(\mu <\lambda \) we obtain

$$\begin{aligned}&-\Delta _p u_\mu -\Delta _q u_\mu +{\hat{\xi }}_\rho u_\mu ^{p-1}-\mu u_\mu ^{-\eta }\nonumber \\&\quad =\mu a(x) u_\mu ^{\tau -1}+f(x,u_\mu )+{\hat{\xi }}_\rho u_\mu ^{p-1}\nonumber \\&\quad =\lambda a(x) u_\mu ^{\tau -1} +f(x,u_\mu )+{\hat{\xi }}_\rho u_\mu ^{p-1} -(\lambda -\mu )a(x)u_\mu ^{\tau -1}\nonumber \\&\quad \le \lambda a(x) u_\lambda ^{\tau -1} +f(x,u_\lambda )+{\hat{\xi }}_\rho u_\lambda ^{p-1}\nonumber \\&\quad \le -\Delta _p u_\lambda -\Delta _q u_\lambda +{\hat{\xi }}_\rho u_\lambda ^{p-1}-\mu u_\lambda ^{-\eta }. \end{aligned}$$
(3.38)

We have

$$\begin{aligned} 0\prec (\lambda -\mu )a(x)u_\mu ^{\tau -1}. \end{aligned}$$

Therefore, from (3.38) and Papageorgiou–Smyrlis [18, Proposition 4], see also Proposition 7 in Papageorgiou–Rădulescu–Repovš [27, Proposition 3.2], we have

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

\(\square \)

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

Proposition 3.10

If hypotheses H(a) and H(f) hold, then \(\lambda ^*<\infty \).

Proof

From hypotheses H(a) and H(f) we can find \({\tilde{\lambda }}>0\) such that

$$\begin{aligned} {\tilde{\lambda }} a(x) s^{\tau -1}+f(x,s) \ge s^{p-1} \quad \text {for a.a. }x\in \Omega \text { and for all }s\ge 0. \end{aligned}$$
(3.39)

Let \(\lambda >{\tilde{\lambda }}\) and suppose that \(\lambda \in {\mathcal {L}}\). Then we can find \(u_\lambda \in {\mathcal {S}}_\lambda \subseteq {{\,\mathrm{int}\,}}\left( C^1_0(\overline{\Omega })_+\right) \). Consider a domain \(\Omega _0\subset \subset \Omega \), that is, \(\Omega _0\subseteq \Omega \) and \(\overline{\Omega }_0\subseteq \Omega \), with a \(C^2\)-boundary \(\partial \Omega _0\) and let \(m_0=\min _{\overline{\Omega }_0}u_\lambda >0\). We set

$$\begin{aligned} m_0^\delta =m_0+\delta \quad \text {with}\quad \delta \in (0,1]. \end{aligned}$$

Let \(\rho =\max \{ \Vert u_\lambda \Vert _\infty , m_0^1\}\) and let \({\hat{\xi }}_\rho >0\) be as postulated by hypothesis H(f)(v). Applying (3.39), hypothesis H(f)(v) and recalling that \(u_\lambda \in \mathcal {S}_\lambda \) as well as \({\tilde{\lambda }}<\lambda \), we obtain

$$\begin{aligned}&-\Delta _p m_0^\delta -\Delta _q m_0^\delta +{\hat{\xi }}_\rho \left( m_0^\delta \right) ^{p-1}-{\tilde{\lambda }} \left( m_0^\delta \right) ^{-\eta }\nonumber \\&\quad \le {\hat{\xi }}_\rho m_0^{p-1}+\chi (\delta ) \quad \text {with } \chi (\delta )\rightarrow 0^+\text { as }\delta \rightarrow 0^+\nonumber \\&\quad \le \left[ {\hat{\xi }}_\rho +1\right] m_0^{p-1}+\chi (\delta )\nonumber \\&\quad \le {\tilde{\lambda }} a(x) m_0^{\tau -1}+f(x,u_0)+{\hat{\xi }}_\rho m_0^{p-1}+\chi (\delta )\nonumber \\&\quad =\lambda a(x) m_0^{\tau -1}+f(x,m_0)+{\hat{\xi }}_\rho m_0^{p-1} -\left( \lambda -{\tilde{\lambda }}\right) m_0^{\tau -1} +\chi (\delta )\nonumber \\&\quad \le \lambda a(x) m_0^{\tau -1} +f(x,m_0)+{\hat{\xi }}_\rho m_0^{p-1} \quad \text {for }\delta \in (0,1]\text { small enough}\nonumber \\&\quad \le \lambda a(x) u_\lambda ^{\tau -1}+f(x,u_\lambda ) +{\hat{\xi }}_\rho u_\lambda ^{p-1}\nonumber \\&\quad =-\Delta _p u_\lambda -\Delta _q u_\lambda +{\hat{\xi }}_\rho u_\lambda ^{p-1} -\lambda u_\lambda ^{-\eta }\nonumber \\&\quad \le -\Delta _p u_\lambda -\Delta _q u_\lambda +{\hat{\xi }}_\rho u_\lambda ^{p-1}-{\tilde{\lambda }} u_\lambda ^{-\eta }\quad \text {for a.\,a.\,}x\in \Omega _0. \end{aligned}$$
(3.40)

From (3.40) and Papageorgiou–Rădulescu–Repovš [27, Proposition 6] we know that

$$\begin{aligned} u_\lambda -m_0^\delta \in D_+ \quad \text {for }\delta \in (0,1] \text { small enough}, \end{aligned}$$

a contradiction. Therefore, \(\lambda ^*\le {\tilde{\lambda }}<\infty \). \(\square \)

Proposition 3.11

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

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

Proof

Let \(\vartheta \in \left( \lambda ,\lambda ^*\right) \). According to Proposition 3.9 we can find \(u_\vartheta \in {\mathcal {S}}_\vartheta \subseteq {{\,\mathrm{int}\,}}\left( C^1_0(\overline{\Omega })_+\right) \) and \(u_0\in {\mathcal {S}}_\lambda \subseteq {{\,\mathrm{int}\,}}\left( C^1_0(\overline{\Omega })_+\right) \) such that

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

Recall that \({\tilde{u}}_\lambda \le u_0\), see Proposition 3.4. Hence \(u_0^{-\eta } \in L^{s}(\Omega )\) for all \(s>N\), see (3.1).

We introduce the Carathéodory function \(i_\lambda :\Omega \times \mathbb {R}\rightarrow \mathbb {R}\) defined by

$$\begin{aligned} i_\lambda (x,s)= {\left\{ \begin{array}{ll} \lambda \left[ u_0(x)^{-\eta }+a(x)u_0(x)^{\tau -1}\right] +f(x,u_0(x))&{}\text {if }s\le u_0(x),\\ \lambda \left[ s^{-\eta }+a(x)s^{\tau -1}\right] +f(x,s)&{}\text {if }u_0(x)<s. \end{array}\right. } \end{aligned}$$
(3.41)

We set \(I_\lambda (x,s)=\int ^s_0 i_\lambda (x,t)\,dt\) and consider the \(C^1\)-functional \(w_\lambda :W^{1,p}_0(\Omega )\rightarrow \mathbb {R}\) defined by

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

Using (3.41) and the nonlinear regularity theory along with the nonlinear maximum principle we can easily check that

$$\begin{aligned} K_{w_\lambda }\subseteq [u_0)\cap {{\,\mathrm{int}\,}}\left( C^1_0(\overline{\Omega })_+\right) . \end{aligned}$$
(3.42)

Then, from (3.41) and (3.42) it follows that, without any loss of generality, we may assume

$$\begin{aligned} K_{w_\lambda }\cap \left[ u_0,u_\vartheta \right] =\{u_0\}. \end{aligned}$$
(3.43)

Otherwise, on account of (3.41) and (3.42), we see that we already have a second positive smooth solution of (P\(_\lambda \)) distinct and larger than \(u_0\).

We introduce the following truncation of \(i_\lambda (x,\cdot )\), namely, \(\hat{i}_\lambda :\Omega \times \mathbb {R}\rightarrow \mathbb {R}\) defined by

$$\begin{aligned} {\hat{i}}_\lambda (x,s)= {\left\{ \begin{array}{ll} i_\lambda (x,s) &{}\text {if }s\le u_\vartheta (x),\\ i_\lambda (x,u_\vartheta (x)) &{}\text {if }u_\vartheta (x)<s, \end{array}\right. } \end{aligned}$$
(3.44)

which is a Carathéodory function. We set \({\hat{I}}_\lambda (x,s)=\int ^s_0 {\hat{i}}_\lambda (x,t)\,dt\) and consider the \(C^1\)-functional \({\hat{w}}_\lambda :W^{1,p}_0(\Omega )\rightarrow \mathbb {R}\) defined by

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

From (3.41) and (3.44) it is clear that \({\hat{w}}_\lambda \) is coercive and due to the Sobolev embedding theorem we know that \({\hat{w}}_\lambda \) is also sequentially weakly lower semicontinuous. Hence, we find \({\hat{u}}_0\in W^{1,p}_0(\Omega )\) such that

$$\begin{aligned} {\hat{w}}_\lambda \left( {\hat{u}}_0\right) =\min \left[ {\hat{w}}_\lambda (u):u\in W^{1,p}_0(\Omega )\right] . \end{aligned}$$
(3.45)

It is easy to see, using (3.44), that

$$\begin{aligned} K_{{\hat{w}}_\lambda }\subseteq \left[ u_0,u_\vartheta \right] \cap {{\,\mathrm{int}\,}}\left( C^1_0(\overline{\Omega })_+\right) \end{aligned}$$
(3.46)

and

$$\begin{aligned} {\hat{w}}_\lambda \big |_{\left[ 0,u_\vartheta \right] } =w_\lambda \big |_{\left[ 0,u_\vartheta \right] }, \quad {\hat{w}}'_\lambda \big |_{\left[ 0,u_\vartheta \right] } =w'_\lambda \big |_{\left[ 0,u_\vartheta \right] }. \end{aligned}$$
(3.47)

From (3.45) we have \({\hat{u}}_0\in K_{{\hat{w}}'_\lambda }\) which by (3.43), (3.46) and (3.47) implies that \({\hat{u}}_0=u_0\).

Recall that \(u_\vartheta -u_0\in {{\,\mathrm{int}\,}}\left( C^1_0(\overline{\Omega })_+\right) \). So, on account of (3.47), we have that \(u_0\) is a local \(C^1_0(\overline{\Omega })\)-minimizer of \(w_\lambda \) and then \(u_0\) is also a local \(W^{1,p}_0(\Omega )\)-minimizer of \(w_\lambda \), see, for example Gasiński–Papageorgiou [7].

We may assume that \(K_{w_\lambda }\) is finite, otherwise, we see from (3.42) that we already have an infinite number of positive smooth solutions of (P\(_\lambda \)) larger than \(u_0\) and so we are done. From Papageorgiou–Rădulescu–Repovš [24, Theorem 5.7.6, p. 449] we find \(\rho \in (0,1)\) small enough such that

$$\begin{aligned} w_\lambda (u_0)<\inf \left[ w_\lambda (u):\Vert u-u_0\Vert =\rho \right] =m_\lambda . \end{aligned}$$
(3.48)

If \(u\in {{\,\mathrm{int}\,}}\left( C^1_0(\overline{\Omega })_+\right) \), then by hypothesis H(f)(ii) we have

$$\begin{aligned} w_\lambda (tu)\rightarrow -\infty \quad \text {as }t\rightarrow +\infty . \end{aligned}$$
(3.49)

Moreover, reasoning as in the proof of Proposition 3.1, we show that

$$\begin{aligned} w_\lambda \text { satisfies the C-condition}, \end{aligned}$$
(3.50)

see also (3.41). Then, (3.48), (3.49) and (3.50) permit the use of the mountain pass theorem. So we can find \({\hat{u}}\in W^{1,p}_0(\Omega )\) such that

$$\begin{aligned} {\hat{u}}\in K_{w_\lambda }\subseteq [u_0)\cap {{\,\mathrm{int}\,}}\left( C^1_0(\overline{\Omega })_+\right) , \quad m_\lambda \le w_\lambda \left( {\hat{u}}\right) . \end{aligned}$$
(3.51)

From (3.51), (3.48) and (3.41) it follows that

$$\begin{aligned} {\hat{u}}\in {\mathcal {S}}_\lambda , \quad u_0\le {\hat{u}}, \quad u_0\ne \hat{u}. \end{aligned}$$

\(\square \)

Remark 3.12

If \(1<q = 2\le \lambda <p\), then, using the tangency principle of Pucci–Serrin [29, p. 35] we can say that \({\hat{u}}-u_0\in {{\,\mathrm{int}\,}}\left( C^1_0(\overline{\Omega })_+\right) \).

Proposition 3.13

If hypotheses H(a) and H(f) hold, then \(\lambda ^*\in {\mathcal {L}}\).

Proof

Let \(\lambda _n\nearrow \lambda ^*\). With \({\hat{u}}_{n+1}\in {\mathcal {S}}_{\lambda _{n+1}}\subseteq {{\,\mathrm{int}\,}}\left( C^1_0(\overline{\Omega })_+\right) \) we introduce the following Carathéodory function (recall that \({\tilde{u}}_{\lambda _1}\le {\tilde{u}}_{\lambda _n}\le u\) for all \(u\in {\mathcal {S}}_{\lambda _n}\) and for all \(n\in \mathbb {N}\), see Propositions 3.4 and 3.7)

$$\begin{aligned}&{\tilde{t}}_n(x,s)=\\&{\left\{ \begin{array}{ll} \lambda _n\left[ {\tilde{u}}_{\lambda _1}(x)^{-\eta }+a(x){\tilde{u}}_{\lambda _1}(x)^{\tau -1} \right] +f\left( x,{\tilde{u}}_{\lambda _1}(x)\right) &{}\text {if } s<{\tilde{u}}_{\lambda _1}(x)\\ \lambda _n\left[ s^{-\eta }+a(x)s^{\tau -1}\right] +f\left( x,s\right) &{}\text {if } {\tilde{u}}_{\lambda _1}(x)\le s \le {\hat{u}}_{n+1}(x)\\ \lambda _n\left[ {\hat{u}}_{n+1}(x)^{-\eta }+a(x){\hat{u}}_{n+1}(x)^{\tau -1}\right] +f\left( x,{\hat{u}}_{n+1}(x)\right) &{}\text {if } {\hat{u}}_{n+1}(x)<s. \end{array}\right. } \end{aligned}$$

Let \({\tilde{T}}_n(x,s)=\int ^s_0 {\tilde{t}}_n(x,t)\,dt\) and consider the \(C^1\)-functional \({\tilde{I}}_n:W^{1,p}_0(\Omega )\rightarrow \mathbb {R}\) defined by

$$\begin{aligned} {\tilde{I}}_n(u)= \frac{1}{p}\Vert \nabla u\Vert _p^p+\frac{1}{q}\Vert \nabla u\Vert _q^q-\int _{\Omega }{\tilde{T}}_n(x,u)\,dx\quad \text {for all }u \in W^{1,p}_0(\Omega ). \end{aligned}$$

Applying the direct method of the calculus of variations, see the definition of the truncation \({\tilde{t}}_n:\Omega \times \mathbb {R}\rightarrow \mathbb {R}\), we can find \(u_n\in W^{1,p}_0(\Omega )\) such that

$$\begin{aligned} {\tilde{I}}_n(u_n)=\min \left[ {\tilde{I}}_n(u):u\in W^{1,p}_0(\Omega )\right] . \end{aligned}$$

Hence, \(\tilde{I}_n'(u_n)=0\) and so \(u_n \in \left[ \tilde{u}_{\lambda _1}, \hat{u}_{n+1}\right] \cap {{\,\mathrm{int}\,}}\left( C^1_0(\overline{\Omega })_+\right) \), see the definition of \(\tilde{t}_n\). Moreover, \(u_n\in \mathcal {S}_{\lambda _n}\subseteq {{\,\mathrm{int}\,}}\left( C^1_0(\overline{\Omega })_+\right) \). From Proposition 2.3 we know that

$$\begin{aligned} {\tilde{I}}_n(u_n)\le {\tilde{I}}_n\left( {\tilde{u}}_{\lambda _1}\right) <0. \end{aligned}$$

Now we introduce the truncation function \({\hat{t}}_n:\Omega \times \mathbb {R}\rightarrow \mathbb {R}\) defined by

$$\begin{aligned} {\hat{t}}_n(x,s)= {\left\{ \begin{array}{ll} \lambda _n\left[ {\tilde{u}}_{\lambda _1}(x)^{-\eta }+a(x){\tilde{u}}_{\lambda _1} (x)^{\tau -1}\right] +f\left( x,{\tilde{u}}_{\lambda _1}(x)\right) &{}\text {if } s\le {\tilde{u}}_{\lambda _1}(x),\\ \lambda _n\left[ s^{-\eta }+a(x)s^{\tau -1}\right] +f(x,s)&{}\text {if } {\tilde{u}}_{\lambda _1}(x)<s. \end{array}\right. } \end{aligned}$$
(3.52)

We set \({\hat{T}}_n(x,s)=\int ^s_0 {\hat{t}}_n(x,t)\,dt\) and consider the \(C^1\)-functional \({\hat{I}}_n:W^{1,p}_0(\Omega )\rightarrow \mathbb {R}\) defined by

$$\begin{aligned} {\hat{I}}_n(u)=\frac{1}{p}\Vert \nabla u\Vert _p^p+\frac{1}{q}\Vert \nabla u\Vert _q^q-\int _{\Omega }{\hat{T}}_n(x,u)\,dx\quad \text {for all }u\in W^{1,p}_0(\Omega ). \end{aligned}$$

It is clear from the definition of the truncation \({\tilde{t}}_n:\Omega \times \mathbb {R}\rightarrow \mathbb {R}\) and (3.52) that

$$\begin{aligned} {\hat{I}}_n\big |_{\left[ 0,{\hat{u}}_{n+1}\right] } ={\tilde{I}}_n\big |_{\left[ 0,{\hat{u}}_{n+1}\right] } \quad \text {and}\quad {\hat{I}}'_n\big |_{\left[ 0,{\hat{u}}_{n+1}\right] } ={\tilde{I}}'_n\big |_{\left[ 0,{\hat{u}}_{n+1}\right] }. \end{aligned}$$

Then from the first part of the proof, we see that we can find a sequence \(u_n\in {\mathcal {S}}_{\lambda _n}\subseteq {{\,\mathrm{int}\,}}\left( C^1_0(\overline{\Omega })_+\right) \), \(n\in \mathbb {N}\), such that

$$\begin{aligned} {\hat{I}}_{n}(u_n)<0\quad \text {for all }n\in \mathbb {N}. \end{aligned}$$
(3.53)

Moreover we have

$$\begin{aligned} \langle {\hat{I}}_n'(u_n),h\rangle =0\quad \text {for all }h\in W^{1,p}_0(\Omega )\text { and for all }n\in \mathbb {N}. \end{aligned}$$
(3.54)

From (3.53) and (3.54), reasoning as in the proof of Proposition 3.1, we show that

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

So we may assume that

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

As before, see the proof of Proposition 3.1, using Proposition 2.1 we show that

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

Then \(u^* \in {\mathcal {S}}_{\lambda ^*} \subseteq {{\,\mathrm{int}\,}}\left( C^1_0(\overline{\Omega })_+\right) \), recall that \({\tilde{u}}_{\lambda _1} \le u_n\) for all \(n\in \mathbb {N}\). This shows that \(\lambda ^*\in {\mathcal {L}}\). \(\square \)

According to Proposition 3.13 we have

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

The set \({\mathcal {S}}_\lambda \) is downward directed, see Papageorgiou–Rădulescu–Repovš [27, Proposition 18] that is, if \(u, {\hat{u}}\in {\mathcal {S}}_\lambda \), we can find \({\tilde{u}}\in {\mathcal {S}}_\lambda \) such that \({\tilde{u}} \le u\) and \({\tilde{u}}\le {\hat{u}}\). Using this fact we can show that, for every \(\lambda \in {\mathcal {L}}\), problem (P\(_\lambda \)) has a smallest positive solution.

Proposition 3.14

If hypotheses H(a) and H(f) hold and if \(\lambda \in {\mathcal {L}}=(0,\lambda ^*]\), then problem (P\(_\lambda \)) has a smallest positive solution \(u_\lambda ^*\in {{\,\mathrm{int}\,}}\left( C^1_0(\overline{\Omega })_+\right) \).

Proof

Applying Lemma 3.10 of Hu–Papageorgiou [12, p. 178] we can find a decreasing sequence \(\{u_n\}_{n \ge 1}\subseteq {\mathcal {S}}_\lambda \) such that

$$\begin{aligned} \inf _{n\ge 1} u_n=\inf {\mathcal {S}}_\lambda . \end{aligned}$$

It is clear that \(\{u_n\}_{n\ge 1}\subseteq W^{1,p}_0(\Omega )\) is bounded. Then, applying Proposition 2.1, we obtain

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

Since \({\tilde{u}}_\lambda \le u_n\) for all \(n\in \mathbb {N}\) it holds \(u^*_\lambda \in {\mathcal {S}}_\lambda \) and \(u^*_\lambda =\inf {\mathcal {S}}_\lambda \). \(\square \)

We examine the map \(\lambda \rightarrow u^*_\lambda \) from \({\mathcal {L}}\) into \(C^1_0(\overline{\Omega })\).

Proposition 3.15

If hypotheses H(a) and H(f) hold, then the map \(\lambda \rightarrow u^*_\lambda \) from \({\mathcal {L}}\) into \(C^1_0(\overline{\Omega })\) is

  1. (a)

    strictly increasing, that is, \(0<\mu <\lambda \le \lambda ^*\) implies \(u^*_\lambda -u^*_\mu \in {{\,\mathrm{int}\,}}\left( C^1_0(\overline{\Omega })_+\right) \);

  2. (b)

    left continuous.

Proof

(a) Let \(0<\mu <\lambda \le \lambda ^*\) and let \(u^*_\lambda \in {{\,\mathrm{int}\,}}\left( C^1_0(\overline{\Omega })_+\right) \) be the minimal positive solution of problem (P\(_\lambda \)), see Proposition 3.14. According to Proposition 3.9 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 \(u^*_\mu \le u_\mu \) we have \(u^*_\lambda -u^*_\mu \in {{\,\mathrm{int}\,}}\left( C^1_0(\overline{\Omega })_+\right) \) and so, we have proved that \(\lambda \rightarrow u^*_\lambda \) is strictly increasing.

(b) Let \(\{\lambda _n\}_{n \ge 1}\subseteq \mathcal {L}=(0,\lambda ^*]\) be such that \(\lambda _n \nearrow \lambda \) as \(n\rightarrow \infty \). We have

$$\begin{aligned} {\tilde{u}}_{\lambda _1} \le u^*_{\lambda _1} \le u^*_{\lambda _n}\le u^*_{\lambda ^*} \quad \text {for all }n \in \mathbb {N}. \end{aligned}$$

Thus,

$$\begin{aligned} \{u^*_{\lambda _n}\}_{n\ge 1}\subseteq W^{1,p}_0(\Omega ) \text { is bounded} \end{aligned}$$

and so

$$\begin{aligned} \{u^*_{\lambda _n}\}_{n\ge 1}\subseteq L^{\infty }(\Omega )\text { is bounded}, \end{aligned}$$

see Guedda–Véron [10, Proposition 1.3]. Therefore, we can find \(\beta \in (0,1)\) and \(c_{19}>0\) such that

$$\begin{aligned} u^*_{\lambda _n} \in C^{1,\beta }_0(\overline{\Omega })\quad \text {and}\quad \Vert u^*_{\lambda _n}\Vert _{C^{1,\beta }_0(\overline{\Omega })}\le c_{19} \quad \text {for all }n\in \mathbb {N}, \end{aligned}$$

see Lieberman [15]. The compact embedding of \(C^{1,\beta }_0(\overline{\Omega })\) into \(C^{1}_0(\overline{\Omega })\) and the monotonicity of \(\{u^*_{\lambda _n}\}_{n\ge 1}\), see part (a), imply that

$$\begin{aligned} u^*_{\lambda _n}\rightarrow \hat{u}^*_{\lambda } \quad \text {in }C^{1}_0(\overline{\Omega }). \end{aligned}$$
(3.55)

If \({\hat{u}}^*_\lambda \ne u^*_\lambda \), then there exists \(x_0\in \Omega \) such that

$$\begin{aligned} u^*_\lambda (x_0)<\hat{u}^*_\lambda (x_0) \quad \text {for all }n\in \mathbb {N}. \end{aligned}$$

From (3.55) we then conclude that

$$\begin{aligned} u^*_\lambda (x_0)<\hat{u}^*_{\lambda _n}(x_0)\quad \text {for all }n\in \mathbb {N}, \end{aligned}$$

which contradicts part (a). Therefore, \({\hat{u}}^*_\lambda = u^*_\lambda \) and so we have proved the left continuity of \(\lambda \rightarrow u^*_\lambda \). \(\square \)

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Papageorgiou, N.S., Winkert, P. (pq)-Equations with Singular and Concave Convex Nonlinearities. Appl Math Optim (2020). https://doi.org/10.1007/s00245-020-09720-0

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Keywords

  • Singular and concave-convex terms
  • Nonlinear regularity theory
  • Nonlinear maximum principle
  • Strong comparison theorems
  • Minimal positive solution

Mathematics Subject Classification

  • Primary: 35J20
  • Secondary: 35J75
  • 35J92