On critical double phase Kirchhoff problems with singular nonlinearity

Abstract

The paper deals with the following double phase problem

$$\begin{aligned} \begin{aligned}&-m \left[ \int _\Omega \left( \frac{|\nabla u|^p}{p} + a(x) \frac{|\nabla u|^q}{q}\right) \,\mathrm {d}x\right] {\text{div}} \left( |\nabla u|^{p-2}\nabla u + a(x) |\nabla u|^{q-2}\nabla u \right) \\&\quad = \lambda u^{-\gamma } +u^{p^*-1}&\quad \text {in } \Omega ,\\&u > 0&\quad \text {in } \Omega ,\\&u = 0&\quad \text {on } \partial \Omega , \end{aligned} \end{aligned}$$

where $\Omega \subset {\mathbb {R}}^N$ is a bounded domain with Lipschitz boundary $\partial \Omega$, $N\ge 2$, m represents a Kirchhoff coefficient, $1<p<q<p^*$ with $p^*=Np/(N-p)$ being the critical Sobolev exponent to p, a bounded weight $a(\cdot )\ge 0$, $\lambda >0$ and $\gamma \in (0,1)$. By the Nehari manifold approach, we establish the existence of at least one weak solution.

1 Introduction

In this paper, we combine the effects of a nonlocal Kirchhoff coefficient and a double phase operator with a singular term and a critical Sobolev nonlinearity. Precisely, we study the problem

where along the paper, and without further mentioning, $\Omega \subset {\mathbb {R}}^N$ is a bounded domain with Lipschitz boundary $\partial \Omega$, dimension $N\ge 2$, $\lambda >0$ is a real parameter and exponent $\gamma \in (0,1)$. The main operator $\mathcal {L}_{p,q}^{a}$ is the so-called double phase operator given by

$$\begin{aligned} \mathcal {L}_{p,q}^{a}(u):= {\text{div}} \big (|\nabla u|^{p-2}\nabla u + a(x) |\nabla u|^{q-2}\nabla u \big ), \quad u\in W^{1,\mathcal {H}}_0(\Omega ), \end{aligned}$$

(1.1)

with $W^{1,\mathcal {H}}_0(\Omega )$ being the homogeneous Musielak-Orlicz Sobolev space where we assume that

($\hbox {h}_1$):

$1<p<N$, $p<q<p^*$ and $0 \le a(\cdot )\in L^\infty (\Omega )$ with $p^*$ being the critical Sobolev exponent to p given by

$$\begin{aligned} p^*=\frac{Np}{N-p}. \end{aligned}$$

(1.2)

While the nonlocal term m in ($P_\lambda$) denotes a Kirchhoff coefficient satisfying

($\hbox {h}_2$):

$m:[0,\infty ) \rightarrow [0,\infty )$ is a continuous function defined by

$$\begin{aligned} m(t) = a_0 + b_0 t^{\theta -1} \quad \text {for all } t\ge 0, \end{aligned}$$

where $a_0\ge 0$, $b_0>0$ with $\theta \in \left[ 1,p^*/q\right)$.

Problem ($P_\lambda$) is said to be of double phase type because of the presence of two different elliptic growths p and q. The study of double phase problems and related functionals originates from the seminal paper by Zhikov [25], where he introduced for the first time in literature the related energy functional to (1.1) defined by

$$\begin{aligned} \omega \mapsto \int _\Omega \big (|\nabla \omega |^p+a(x)|\nabla \omega |^q\big )\,\mathrm {d}x. \end{aligned}$$

(1.3)

This kind of functional has been used to describe models for strongly anisotropic materials in the context of homogenization and elasticity. Indeed, the modulating coefficient $a(\cdot )$ dictates the geometry of composites made of two different materials with distinct power hardening exponents p and q. From the mathematical point of view, the behavior of (1.3) is related to the sets on which the weight function $a(\cdot )$ vanishes or not. In this direction, Zhikov found other mathematical applications for (1.3) in the study of duality theory and of the Lavrentiev gap phenomenon, as shown in [26, 27]. Also, (1.3) belongs to the class of the integral functionals with nonstandard growth condition, according to Marcellini’s terminology [22, 23]. Following this line of research, Mingione et al. provide famous results in the regularity theory of local minimizers of (1.3), see, for example, the works of Baroni-Colombo-Mingione [4, 5] and Colombo-Mingione [9, 10].

Starting from [25], several authors studied existence and multiplicity results for nonlinear problems driven by (1.1) with the help of different variational techniques. In particular, Fiscella-Pinamonti [18] introduced two different double phase problems of Kirchhoff type, with the same variational structure set in $W^{1,\mathcal {H}}_0(\Omega )$. By the mountain pass and fountain theorems, existence and multiplicity results are provided in [18]. Following this direction, in [17] Fiscella-Marino-Pinamonti-Verzellesi consider some classes of Kirchhoff type problems on a double phase setting but with nonlinear boundary conditions. Combining variational methods, truncation arguments and topological tools, different multiplicity results are established. Recently, the authors [2] were able to study a Kirchhoff problem like ($P_\lambda$), but involving a subcritical term. By a suitable Nehari manifold decomposition, the existence of two different solutions are provided in [2]. We also mention the works of Cammaroto-Vilasi [7], Isernia-Repovš [20] and Ambrosio-Isernia [1] for Kirchhoff type problems driven by the $p(\cdot )$-Laplacian or the (p, q)-Laplacian.

The main novelty, as well as the main difficulty, of problem ($P_\lambda$) is the presence of a critical Sobolev nonlinearity. Indeed, in order to overcome the lack of compactness at critical levels arising from the presence of the critical term in ($P_\lambda$), the same fibering analysis used in [2] cannot work. For this, we exploit other variational tools inspired by more recent situations as in [14]. For this, Farkas-Fiscella-Winkert [14] used a suitable convergence analysis of gradients in order to handle the critical Sobolev nonlinearity of problem

$$\begin{aligned} \begin{aligned} -\text{div} \left( |\nabla u|^{p-2}\nabla u + a(x) |\nabla u|^{q-2}\nabla u \right)&=\lambda |u|^{\vartheta -2}u+|u|^{p^*-2}u&\text {in } \Omega , \\ u&= 0&\text {on } \partial \Omega . \end{aligned} \end{aligned}$$

Following this direction, we mention [15, 16] concerning existence results for critical double phase problems involving a singular term and defined on Minkowski spaces in terms of Finsler manifolds, that is driven by the Finsler double phase operator

$$\begin{aligned} \mathcal {L}_{p,q}^{F,a}(u):= \text{div} \left( F^{p-1}(\nabla u)\nabla F(\nabla u)+ a(x) F^{q-1}(\nabla u)\nabla F(\nabla u) \right) , \end{aligned}$$

where $({\mathbb {R}}^N,F)$ stands for a Minkowski space. While, Crespo-Blanco-Papageorgiou-Winkert [12] consider a nonhomogeneous singular Neumann double phase problem with critical growth on the boundary, given by

$$\begin{aligned} \begin{aligned} -\text{div} \left( |\nabla u|^{p-2}\nabla u + a(x) |\nabla u|^{q-2}\nabla u \right) +\alpha (x)u^{p-1}&= \zeta (x)u^{-\gamma }+\lambda u^{q_1-1}&\text {in } \Omega , \\ \left( |\nabla u|^{p-2} \nabla u+ a(x) |\nabla u|^{q-2} \nabla u\right) \cdot \nu&= -\beta (x)u^{p_* -1}&\text {on } \partial \Omega . \end{aligned} \end{aligned}$$

(1.4)

By the fibering approach introduced by Drábek-Pohozaev [13] along with a Nehari manifold decomposition, the existence of at least two solutions of (1.4) is obtained in [12].

Inspired by the above papers, we solve problem ($P_\lambda$) by a variational approach. Indeed, a function $u \in W^{1,\mathcal {H}}_0(\Omega )$ is said to be a weak solution of problem ($P_\lambda$) if $u^{-\gamma }\varphi \in L^1(\Omega )$, $u>0$ a.e. in $\Omega$ and

$$\begin{aligned} m(\phi _\mathcal {H}(\nabla u)) \left\langle \mathcal {L}_{p,q}^{a}(u), \varphi \right\rangle = \lambda \int _{\Omega } u^{-\gamma }\varphi \,\mathrm {d}x +\int _\Omega u^{p^*-1} \varphi \,\mathrm {d}x \end{aligned}$$

is satisfied for all $\varphi \in W^{1,\mathcal {H}}_0(\Omega )$, where $\langle \cdot ,\cdot \rangle$ denotes the duality pairing between $W^{1,\mathcal {H}}_0(\Omega )$ and its dual space $W^{1,\mathcal {H}}_0(\Omega )^*$ In particular, the weak solutions of ($P_\lambda$) are the critical points of the energy functional $J_\lambda :W^{1,\mathcal {H}}_0(\Omega )\rightarrow {\mathbb {R}}$ given by

$$\begin{aligned} J_\lambda (u) = \left[ a_0\phi _{\mathcal {H}}(\nabla u)+\frac{b_0}{\theta }\phi _{\mathcal {H}}^\theta (\nabla u)\right] -\frac{\lambda }{1-\gamma }\int _\Omega |u|^{1-\gamma }\,\mathrm {d}x - \frac{1}{p^*}\int _\Omega |u|^{p^*}\,\mathrm {d}x, \end{aligned}$$

for any $u\in W^{1,\mathcal {H}}_0(\Omega )$, where

$$\begin{aligned} \phi _{\mathcal {H}}(u)= \int _\Omega \left( \frac{|u|^p}{p}+a(x) \frac{|u|^q}{q}\right) \,\mathrm {d}x. \end{aligned}$$

Hence, the main result reads as follows.

Theorem 1.1

Let hypotheses ($\hbox {h}_1$)-($\hbox {h}_2$) be satisfied. Then there exists $\lambda ^*>0$ such that for all $\lambda \in (0,\lambda ^*]$ problem ($P_\lambda$) has at least one weak solution $u_\lambda$ such that $J_\lambda (u_\lambda )<0$.

The proof of Theorem 1.1 is based on a suitable minimization argument on the Nehari manifold. For this, we extract a minimizing sequence whose energy values converge to a negative number. However, in order to verify that the sequence actually converges to a solution of ($P_\lambda$) we need a truncation argument combined with a delicate gradient analysis, inspired by [14].

The paper is organized as follows. In Sect. 2, we recall the main properties of Musielak-Orlicz Sobolev spaces $W^{1,\mathcal {H}}_0(\Omega )$ and state the main embeddings concerning these spaces. Section 3 gives a detailed analysis of the fibering map, presents the main properties of suitable subsets of the Nehari manifold and finally shows the existence of a weak solution of problem ($P_\lambda$).

2 Preliminaries

In this section, we will present the main properties and embedding results for Musielak-Orlicz Sobolev spaces. First, we denote by $L^{r}(\Omega )=L^r(\Omega ;{\mathbb {R}})$ and $L^r(\Omega ;{\mathbb {R}}^N)$ the usual Lebesgue spaces with the norm $\Vert \cdot \Vert _r$ and the corresponding Sobolev space $W^{1,r}_0(\Omega )$ is equipped with the norm $\Vert \nabla \cdot \Vert _r$, for $1\le r\le \infty$.

Suppose hypothesis ($\hbox {h}_1$) and consider the nonlinear function $\mathcal {H}:\Omega \times [0,\infty )\rightarrow [0,\infty )$ defined by

$$\begin{aligned} {\mathcal {H}}(x,t)= t^p+a(x)t^q. \end{aligned}$$

The Musielak-Orlicz Lebesgue space $L^\mathcal {H}(\Omega )$ is given by

$$\begin{aligned} L^\mathcal {H}(\Omega )=\left\{ u:\Omega \rightarrow {\mathbb {R}}\,\Big |\,\,u\text { is measurable and }\varrho _{\mathcal {H}}(u)<\infty \right\} \end{aligned}$$

equipped with the Luxemburg norm

$$\begin{aligned} \Vert u\Vert _{\mathcal {H}} = \inf \left\{ \tau >0\,\Big |\, \varrho _{\mathcal {H}}\left( \frac{u}{\tau }\right) \le 1 \right\} , \end{aligned}$$

where the modular function is given by

$$\begin{aligned} \varrho _{\mathcal {H}}(u):=\int _\Omega \mathcal {H}(x,|u|)\,\mathrm {d}x=\int _\Omega \big (|u|^{p}+a(x)|u|^q\big )\,\mathrm {d}x. \end{aligned}$$

Next, we recall the relation between the norm $\Vert \,\cdot \,\Vert _{\mathcal {H}}$ and the modular function $\varrho _{\mathcal {H}}$, see Liu-Dai [21, Proposition 2.1] or Crespo-Blanco-Gasiński-Harjulehto-Winkert [11, Proposition 2.13].

Proposition 2.1

Let ($\hbox {h}_1$) be satisfied, $u\in L^{\mathcal {H}}(\Omega )$ and $c>0$. Then the following hold:

• (i) If $u\ne 0$, then $\Vert u\Vert _{\mathcal {H}}=c$ if and only if $\varrho _{\mathcal {H}}(\frac{u}{c})=1$;

• (ii) $\Vert u\Vert _{\mathcal {H}}<1$ (resp. $>1$, $=1$) if and only if $\varrho _{\mathcal {H}}(u)<1$ (resp. $>1$, $=1$);

• (iii) If $\Vert u\Vert _{\mathcal {H}}<1$, then $\Vert u\Vert _{\mathcal {H}}^q\le \varrho _{\mathcal {H}}(u)\le \Vert u\Vert _{\mathcal {H}}^p$;

• (iv) If $\Vert u\Vert _{\mathcal {H}}>1$, then $\Vert u\Vert _{\mathcal {H}}^p\le \varrho _{\mathcal {H}}(u)\le \Vert u\Vert _{\mathcal {H}}^q$;

• (v) $\Vert u\Vert _{\mathcal {H}}\rightarrow 0$ if and only if $\varrho _{\mathcal {H}}(u)\rightarrow 0$;

• (vi) $\Vert u\Vert _{\mathcal {H}}\rightarrow \infty$ if and only if $\varrho _{\mathcal {H}}(u)\rightarrow \infty$.

Moreover, we define the weighted space

$$\begin{aligned} L^q_a(\Omega )=\left\{ u:\Omega \rightarrow {\mathbb {R}}\,\Big |\,\,u\text { is measurable and }\int _\Omega a(x) |u|^q \,\mathrm {d}x< \infty \right\} \end{aligned}$$

endowed with the seminorm

$$\begin{aligned} \Vert u\Vert _{q,a} = \left( \int _\Omega a(x) |u|^q \,\mathrm {d}x \right) ^{\frac{1}{q}}. \end{aligned}$$

The corresponding Musielak-Orlicz Sobolev space $W^{1,\mathcal {H}}(\Omega )$ is defined by

$$\begin{aligned} W^{1,\mathcal {H}}(\Omega )= \Big \{u \in L^\mathcal {H}(\Omega ) \,:\, |\nabla u| \in L^{\mathcal {H}}(\Omega ) \Big \} \end{aligned}$$

equipped with the norm

$$\begin{aligned} \Vert u\Vert _{1,\mathcal {H}}= \Vert \nabla u \Vert _{\mathcal {H}}+\Vert u\Vert _{\mathcal {H}}, \end{aligned}$$

where $\Vert \nabla u\Vert _\mathcal {H}=\Vert \,|\nabla u|\,\Vert _{\mathcal {H}}$. In addition, we denote by $W^{1,\mathcal {H}}_0(\Omega )$ the completion of $C^\infty _0(\Omega )$ in $W^{1,\mathcal {H}}(\Omega )$. Thanks to hypothesis ($\hbox {h}_1$), we know that

$$\begin{aligned} \Vert u\Vert =\Vert \nabla u\Vert _{\mathcal {H}}, \end{aligned}$$

is an equivalent norm in $W^{1,\mathcal {H}}_0(\Omega )$, see Crespo-Blanco-Gasiński-Harjulehto-Winkert [11, Proposition 2.16(ii)]. Furthermore, it is known that $L^\mathcal {H}(\Omega )$, $W^{1,\mathcal {H}}(\Omega )$ and $W^{1,\mathcal {H}}_0(\Omega )$ are uniformly convex and so reflexive Banach spaces, see Colasuonno-Squassina [8, Proposition 2.14] or Harjulehto-Hästö [19, Theorem 6.1.4].

Finally, we recall some useful embedding results for the spaces $L^\mathcal {H}(\Omega )$ and $W^{1,\mathcal {H}}_0(\Omega )$, see Colasuonno-Squassina [8, Proposition 2.15] or Crespo-Blanco-Gasiński-Harjulehto-Winkert [11, Propositions 2.17 and 2.19].

Proposition 2.2

Let ($\hbox {h}_1$) be satisfied and let $p^*$ be the critical exponent to p given in (1.2). Then the following embeddings hold:

• (i) $L^{\mathcal {H}}(\Omega ) \hookrightarrow L^{r}(\Omega )$ and $W^{1,\mathcal {H}}_0(\Omega )\hookrightarrow W^{1,r}_0(\Omega )$ are continuous for all $r\in [1,p]$;

• (ii) $W^{1,\mathcal {H}}_0(\Omega )\hookrightarrow L^{r}(\Omega )$ is continuous for all $r \in [1,p^*]$ and compact for all $r \in [1,p^*)$;

• (iii) $L^{\mathcal {H}}(\Omega ) \hookrightarrow L^q_a(\Omega )$ is continuous;

• (iv) $L^{q}(\Omega )\hookrightarrow L^{\mathcal {H}}(\Omega )$ is continuous.

3 Proof the main result

In order to solve problem ($P_\lambda$), we apply a minimization argument for $J_\lambda$ on a suitable subset of $W^{1,\mathcal {H}}_0(\Omega )$. For this, we define the fibering function $\psi _u:[0,\infty ) \rightarrow {\mathbb {R}}$ defined by

$$\begin{aligned} \psi _u(t)= J_\lambda (tu)\quad \text {for all }t\ge 0, \end{aligned}$$

which gives

$$\begin{aligned} \psi _u(t) = \left[ a_0\phi _\mathcal {H}(t\nabla u) +\frac{b_0}{\theta }\phi _\mathcal {H}^{\theta }(t\nabla u)\right] - \lambda \frac{t^{1-\gamma }}{1-\gamma }\int _\Omega |u|^{1-\gamma }\,\mathrm {d}x - \frac{t^{p^*}}{p^*}\int _\Omega |u|^{p^*}\,\mathrm {d}x. \end{aligned}$$

It is easy to see that $\psi _u \in C^{\infty }((0,\infty ))$. In particular, we have for $t>0$

$$\begin{aligned} \psi _u'(t)&= \left[ a_0 +b_0\phi _\mathcal {H}^{\theta -1}(t\nabla u)\right] \left( t^{p-1}\Vert \nabla u\Vert _p^p+t^{q-1} \Vert \nabla u\Vert _{q,a}^q\right) \\&\quad - \lambda t^{-\gamma }\int _\Omega |u|^{1-\gamma }\,\mathrm {d}x - t^{p^*-1}\int _\Omega |u|^{p^*}\,\mathrm {d}x \end{aligned}$$

and

$$\begin{aligned} \psi _u''(t)&= \left[ a_0 +b_0\phi _\mathcal {H}^{\theta -1}(t\nabla u)\right] \left[ (p-1)t^{p-2}\Vert \nabla u\Vert _p^p+(q-1)t^{q-2} \Vert \nabla u\Vert _{q,a}^q\right] \\&\qquad + b_0(\theta -1)\phi _\mathcal {H}^{\theta -2}(t\nabla u)\left( t^{p-1}\Vert \nabla u\Vert _p^p+t^{q-1} \Vert \nabla u\Vert _{q,a}^q\right) ^2\\&\qquad + \lambda \gamma t^{-\gamma -1}\int _\Omega |u|^{1-\gamma }\,\mathrm {d}x -(p^*-1) t^{p^*-2}\int _\Omega |u|^{p^*}\,\mathrm {d}x. \end{aligned}$$

Thus, we can introduce the Nehari manifold related to our problem which is defined by

$$\begin{aligned} \mathcal {N}_\lambda = \left\{ u\in W^{1,\mathcal {H}}_0(\Omega )\setminus \{0\}\,:\, \psi _u'(1)=0\right\} . \end{aligned}$$

In particular, we have $u \in \mathcal {N}_\lambda$ if and only if

$$\begin{aligned} \left[ a_0 +b_0\phi _\mathcal {H}^{\theta -1}(\nabla u)\right] \left( \Vert \nabla u\Vert _p^p+ \Vert \nabla u\Vert _{q,a}^q\right) = \lambda \int _\Omega |u|^{1-\gamma }\,\mathrm {d}x + \int _\Omega |u|^{p^*}\,\mathrm {d}x. \end{aligned}$$

Also $tu\in \mathcal {N}_\lambda$ if and only if $\psi _{tu}'(1)=0$. Observe that $\mathcal {N}_\lambda$ contains all weak solutions of ($P_\lambda$). Moreover, we define the following subsets of $\mathcal {N}_\lambda$

$$\begin{aligned} \mathcal {N}_\lambda ^{+} = \left\{ u\in \mathcal {N}_\lambda \,:\, \psi _u''(1)> 0\right\} \quad \text {and}\quad \mathcal {N}_\lambda ^{\circ } = \left\{ u\in \mathcal {N}_\lambda \,:\, \psi _u''(1)= 0\right\} . \end{aligned}$$

In contrast to [2] we are not going to study the set $\mathcal {N}_\lambda ^{-} = \left\{ u\in \mathcal {N}_\lambda \,:\, \psi _u''(1)< 0\right\}$. The next Lemma can be shown as in [2, Lemmas 3.1 and 3.2] replacing r by $p^*$.

Lemma 3.1

Let hypotheses ($\hbox {h}_1$)-($\hbox {h}_2$) be satisfied.

• (i) The functional $J_\lambda \big |_{\mathcal {N}_\lambda }$ is coercive and bounded from below for any $\lambda >0$.

• (ii) There exists $\Lambda _1>0$ such that $\mathcal {N}_\lambda ^\circ =\emptyset$ for all $\lambda \in (0,\Lambda _1)$.

Let S be the best Sobolev constant in $W_0^{1,p}(\Omega )$ defined as

$$\begin{aligned} S= \inf _{{ u\in W^{1,p}_0(\Omega )\setminus \{0\}}}\frac{\Vert \nabla u\Vert ^p_p}{\Vert u\Vert _{p^*}^p}. \end{aligned}$$

(3.1)

Note that we can write $\psi _u'(t)$ in the form

$$\begin{aligned} \psi _u'(t)= t^{-\gamma } \left( \sigma _u(t) -\lambda \int _\Omega |u|^{1-\gamma }\,\mathrm {d}x\right) , \quad t>0, \end{aligned}$$

(3.2)

where

$$\begin{aligned} \sigma _u(t)= \left[ a_0 +b_0\phi _\mathcal {H}^{\theta -1}(t\nabla u)\right] \left( t^{p-1+\gamma }\Vert \nabla u\Vert _p^p+t^{q-1+\gamma } \Vert \nabla u\Vert _{q,a}^q\right) - t^{p^*-1+\gamma }\int _\Omega |u|^{p^*}\,\mathrm {d}x. \end{aligned}$$

From this definition we see that $tu\in \mathcal {N}_\lambda$ if and only if

$$\begin{aligned} \sigma _u(t)= \lambda \int _\Omega |u|^{1-\gamma }\,\mathrm {d}x. \end{aligned}$$

(3.3)

The next Lemma shows that $\mathcal {N}_\lambda ^+$ is nonempty whenever $\lambda$ is sufficiently small.

Lemma 3.2

Let hypotheses ($\hbox {h}_1$)-($\hbox {h}_2$) be satisfied and let $u\in W^{1,\mathcal {H}}_0(\Omega )\setminus \{0\}$. Then there exist $\Lambda _2>0$ and unique $t_1^u<t_{\max }^u<t_2^u$ such that

$$\begin{aligned} 0<\sigma _u'(t_1^u)= (t_1^u)^\gamma \psi _u''(t_1^u), \quad 0>\sigma _u'(t_2^u)= (t_2^u)^\gamma \psi _u''(t_2^u) \quad \text {and}\quad \sigma _u(t_{\max }^u)=\max _{t>0} \sigma _u(t) \end{aligned}$$

whenever $\lambda \in (0,\Lambda _2)$. In particular, $t_1^u u \in \mathcal {N}_\lambda ^+$ for $\lambda \in (0,\Lambda _2)$.

Proof

For $u\in W^{1,\mathcal {H}}_0(\Omega )\setminus \{0\}$ the equation

$$\begin{aligned} 0=\sigma _u'(t)&= \left[ a_0 +b_0\phi _\mathcal {H}^{\theta -1}(t\nabla u)\right] \left[ (p-1+\gamma )t^{p-2+\gamma }\Vert \nabla u\Vert _p^p+ (q-1+\gamma )t^{q-2+\gamma } \Vert \nabla u\Vert _{q,a}^q\right] \\&\qquad + b_0(\theta -1) \phi _\mathcal {H}^{\theta -2}(t\nabla u)\left( t^{p-1+\gamma }\Vert \nabla u\Vert _p^p+t^{q-1+\gamma } \Vert \nabla u\Vert _{q,a}^q\right) \\&\qquad \left( t^{p-1}\Vert \nabla u\Vert _p^p+t^{q-1} \Vert \nabla u\Vert _{q,a}^q\right) \\&\qquad - (p^*-1+\gamma )t^{p^*-2+\gamma }\int _\Omega |u|^{p^*}\,\mathrm {d}x \end{aligned}$$

can be equivalently written as

$$\begin{aligned} \begin{aligned}&\left[ a_0 +b_0\phi _\mathcal {H}^{\theta -1}(t\nabla u)\right] \left[ (p-1+\gamma )t^{p-p^*}\Vert \nabla u\Vert _p^p+ (q-1+\gamma )t^{q-p^*} \Vert \nabla u\Vert _{q,a}^q\right] \\&\quad + b_0(\theta -1) \phi _\mathcal {H}^{\theta -2}(t\nabla u)\left( t^{p-p^*+1}\Vert \nabla u\Vert _p^p+t^{q-p^*+1} \Vert \nabla u\Vert _{q,a}^q\right) \left( t^{p-1}\Vert \nabla u\Vert _p^p+t^{q-1} \Vert \nabla u\Vert _{q,a}^q\right) \\&\quad = (p^*-1+\gamma )\int _\Omega |u|^{p^*}\,\mathrm {d}x. \end{aligned} \end{aligned}$$

(3.4)

From $p^*>q\theta$ and $\theta \ge 1$ we see that

$$\begin{aligned} \begin{aligned} p(\theta -1)+p-p^*&< \min \left\{ p(\theta -1)+q-p^*,q(\theta -1)+p-p^*\right\} \\&\le \max \left\{ p(\theta -1)+q-p^*,q(\theta -1)+p-p^*\right\} \\&<q(\theta -1)+q-p^*=q\theta -p^*<0. \end{aligned} \end{aligned}$$

(3.5)

We denote the left-hand side of (3.4) by

$$\begin{aligned} \begin{aligned} T_u(t)=&\left[ a_0 +b_0\phi _\mathcal {H}^{\theta -1}(t\nabla u)\right] \left[ (p-1+\gamma )t^{p-p^*}\Vert \nabla u\Vert _p^p+ (q-1+\gamma )t^{q-p^*} \Vert \nabla u\Vert _{q,a}^q\right] \\&+ b_0(\theta -1) \phi _\mathcal {H}^{\theta -2}(t\nabla u)\left( t^{p-p^*+1}\Vert \nabla u\Vert _p^p+t^{q-p^*+1} \Vert \nabla u\Vert _{q,a}^q\right) \\&\left( t^{p-1}\Vert \nabla u\Vert _p^p+t^{q-1} \Vert \nabla u\Vert _{q,a}^q\right) . \end{aligned} \end{aligned}$$

Then, from (3.5) and $0<\gamma<1<p<q<p^*$, we know that

$$\begin{aligned} \text {(i)} \quad \lim _{t\rightarrow 0^+} T_u(t)=\infty ,\quad \text {(ii)}\quad \lim _{t\rightarrow \infty } T_u(t)=0,\quad \text {(iii)} \quad T_u'(t)<0 \ \text { for all }t>0. \end{aligned}$$

From the intermediate value theorem along with (i) and (ii) we can find $t_{\max }^u>0$ such that (3.4) holds. In addition, (iii) implies that $t_{\max }^u$ is unique due to the injectivity of $T_u$. Moreover, if we consider $\sigma '_u(t)>0$, then in place of (3.4) we get

$$\begin{aligned} T_u(t)> (p^*-1+\gamma )\int _\Omega |u|^{p^*}\,\mathrm {d}x. \end{aligned}$$

Since $T_u$ is strictly decreasing, this holds for all $t<t_{\max }^u$. The same can be said for $\sigma _u'(t)<0$ and $t>t_{\max }^u$. Hence, $\sigma _u$ is injective in $(0,t_{\max }^u)$ and in $(t_{\max }^u,\infty )$. Furthermore,

$$\begin{aligned} \sigma _u(t_{\max }^u)=\max _{t>0} \sigma _u(t) \end{aligned}$$

with the global maximum $t_{\max }^u>0$ of $\sigma _u$. Moreover, we have

$$\begin{aligned} \lim _{t\rightarrow 0^+} \sigma _u(t) = 0 \quad \text {and}\quad \lim _{t\rightarrow \infty } \sigma _u(t) = -\infty . \end{aligned}$$

Applying the estimate $p \phi _{\mathcal {H}}(\nabla u) \ge \Vert \nabla u\Vert _p^p$ we obtain

$$\begin{aligned} \sigma _u'(t) \ge \frac{b_0}{p^{\theta -1}}(p\theta -1+\gamma )t^{p\theta -2+\gamma }\Vert \nabla u\Vert _p^{p\theta } - (p^*-1+\gamma )t^{p^*-2+\gamma }\int _\Omega |u|^{p^*}\,\mathrm {d}x, \end{aligned}$$

(3.6)

which by using Hölder’s inequality and (3.1) results in

$$\begin{aligned} t_{\max }^u \ge \frac{1}{\Vert \nabla u\Vert _p}\left( \frac{b_0(p\theta -1+\gamma )S^{\frac{p^*}{p}}}{p^{\theta -1}(p^*-1+\gamma )}\right) ^{\frac{1}{p^*-p\theta }} := t_0^{u}. \end{aligned}$$

(3.7)

Note that $\sigma _u$ is increasing on $(0,t_{\max }^u)$. Hence from $p \phi _{\mathcal {H}}(\nabla u) \ge \Vert \nabla u\Vert _p^p$, $p<q$, Hölder’s inequality, (3.1) and the representation of $t_0^u$ in (3.7) we have

$$\begin{aligned} \sigma _u(t_{\max }^u)&\ge \sigma _u(t_0^u) \ge \frac{b_0}{p^{\theta -1}} (t_0^u)^{p \theta -1+\gamma }\Vert \nabla u\Vert _p^{p\theta } - (t_0^u)^{p^*-1+\gamma }\int _\Omega |u|^{p^*}\,\mathrm {d}x\\&\ge (t_0^u)^{p \theta -1+\gamma } \Vert \nabla u\Vert _p^{p\theta } \left( \frac{b_0}{p^{\theta -1}}- (t_0^u)^{p^*-p\theta } S^{\frac{-p^*}{p}} \Vert \nabla u\Vert ^{p^*-p\theta }_p\right) \\&\ge \left( \frac{p^*-p \theta }{p^*-1+\gamma }\right) \frac{b_0}{p^{\theta -1}} (t_0^u)^{p \theta -1+\gamma } \Vert \nabla u\Vert _p^{p\theta }\\ {}&> \left( \frac{p^*-q \theta }{p^*-1+\gamma }\right) \frac{b_0}{p^{\theta -1}} (t_0^u)^{p \theta -1+\gamma } \Vert \nabla u\Vert _p^{p\theta } \\&= \left( \frac{p^*-q\theta }{p^*-1+\gamma }\right) \Vert \nabla u\Vert ^{1-\gamma }_p\frac{b_0}{p^{\theta -1}}\left( \frac{b_0(p\theta -1+\gamma )S^{\frac{p^*}{p}}}{p^{\theta -1}(p^*-1+\gamma )}\right) ^{\frac{p\theta -1+\gamma }{p^*-p\theta }}\\&\ge \Lambda _2 \int _\Omega |u|^{1-\gamma }\,\mathrm {d}x, \end{aligned}$$

where $\Lambda _2$ is given by

$$\begin{aligned} \Lambda _2=\frac{b_0}{p^{\theta -1}} \left( \frac{p^*-q \theta }{p^*-1+\gamma }\right) \left( \frac{b_0(p\theta -1+\gamma )S}{p^{\theta -1}(p^*-1+\gamma )}\right) ^{\frac{p\theta -1+\gamma }{p^*-p\theta }} \frac{S^{\frac{1-\gamma }{p}}}{|\Omega |^{\frac{p^*+\gamma -1}{p^*}}}. \end{aligned}$$

From the considerations above we conclude that

$$\begin{aligned} \sigma _u(t_{\max }^u) > \lambda \int _\Omega |u|^{1-\gamma }\,\mathrm {d}x \end{aligned}$$

whenever $\lambda \in (0,\Lambda _2)$. Since $\sigma _u$ is injective in $(0,t_{\max }^u)$ and in $(t_{\max }^u,\infty )$, we can find unique $t_1^u, t_2^u>0$ such that

$$\begin{aligned} \sigma _u(t_1^u) = \lambda \int _\Omega |u|^{1-\gamma }\,\mathrm {d}x = \sigma _u(t_2^u) \quad \text {with}\quad \sigma _u'(t_2^u)<0<\sigma _u'(t_1^u). \end{aligned}$$

Due to (3.3) we have $t_1^u u\in \mathcal {N}_\lambda$. Then, from the representation in (3.2), we observe that

$$\begin{aligned} \sigma _u'(t) = t^\gamma \psi _u''(t)+\gamma t^{\gamma -1}\psi _u'(t). \end{aligned}$$

Finally, since $\psi _u'(t_1^u)=\psi _u'(t_2^u)=0$ and $\sigma _u'(t_2^u)<0<\sigma _u'(t_1^u)$ we derive that

$$\begin{aligned} 0<\sigma _u'(t_1^u)= (t_1^u)^\gamma \psi _u''(t_1^u) \quad \text {and}\quad 0>\sigma _u'(t_2^u)= (t_2^u)^\gamma \psi _u''(t_2^u). \end{aligned}$$

This shows, in particular, that $t_1^u u \in \mathcal {N}_\lambda ^+$ for $\lambda \in (0,\Lambda _2)$. $\square$

Next we show that the modular $\varrho _\mathcal {H}(\nabla \cdot )$ is upper bounded with respect to the elements of $\mathcal {N}_\lambda ^+$. The proof is similar to that in [2, Proposition 3.4] and so we omitted it.

Lemma 3.3

Let hypotheses ($\hbox {h}_1$)-($\hbox {h}_2$) be satisfied. Then there exist $\Lambda _3>0$ and constant $D_1=D_1(\lambda )>0$ such that

$$\begin{aligned} \varrho _\mathcal {H}(\nabla u)=\Vert \nabla u\Vert _p^p + \Vert \nabla u\Vert _{q,a}^q<D_1 \end{aligned}$$

for every $u\in \mathcal {N}_\lambda ^+$ and for every $\lambda \in (0,\Lambda _3)$.

By Lemma 3.1(ii), we observe that $\mathcal N^+_\lambda$ is closed in $W^{1,\mathcal {H}}_0(\Omega )$ for $\lambda >0$ small enough. We define

$$\begin{aligned} \Theta _\lambda ^+ = \inf _{u\in \mathcal {N}_\lambda ^+}J_\lambda (u). \end{aligned}$$

The next proposition shows that $\Theta _\lambda ^+ <0$. We refer to [2, Proposition 4.1] for its proof.

Proposition 3.4

Let hypotheses ($\hbox {h}_1$)-($\hbox {h}_2$) be satisfied and let $\lambda \in (0,\min \{\Lambda _1,\Lambda _2\})$, with $\Lambda _1$, $\Lambda _2$ given in Lemmas 3.1(ii) and 3.2. Then $\Theta _\lambda ^+<0$.

Based on the implicit function theorem in its version stated in Berger [6, p. 115] we can proof the following Lemma which proof is similar to the one in [2, Lemma 4.2].

Lemma 3.5

Let hypotheses ($\hbox {h}_1$)-($\hbox {h}_2$) be satisfied and let $\lambda >0$. Let us consider $u\in \mathcal {N}_\lambda ^+$. Then there exist $\varepsilon >0$ and a continuous function $\zeta :B_\varepsilon (0) \rightarrow (0,\infty )$ such that

$$\begin{aligned} \zeta (0)=1 \quad \text {and}\quad \zeta (v)(u+v)\in \mathcal {N}_\lambda ^+ \quad \text {for all } v \in B_\varepsilon (0), \end{aligned}$$

where $B_\varepsilon (0) := \{v\in W^{1,\mathcal {H}}_0(\Omega )\,:\, \Vert v\Vert <\varepsilon \}$.

Now, we set $\Lambda ^*:=\min \{\Lambda _1,\Lambda _2,\Lambda _3\}$ with $\Lambda _1$, $\Lambda _2$ and $\Lambda _3>0$ given in Lemmas 3.1(ii), 3.2 and 3.3 . Let $\lambda \in (0,\Lambda ^*)$. Applying Ekeland’s variational principle, we obtain a sequence $\{u_n\}_{n\in {\mathbb {N}}}\subset \mathcal {N}_\lambda ^+$ satisfying

$$\begin{aligned} \theta _\lambda ^+< J_\lambda (u_n) < \theta _\lambda ^+ +\frac{1}{n}, \end{aligned}$$

(3.8)

$$\begin{aligned} J_\lambda (u) \ge J_\lambda (u_n) + \frac{\Vert u-u_n\Vert }{n} \end{aligned}$$

(3.9)

for any $u\in {\mathcal {N}}_\lambda ^+$. By Lemma 3.1(i), we know that $\{u_n\}_{n\in {\mathbb {N}}}$ is bounded in $W^{1,\mathcal {H}}_0(\Omega )$. Hence, by Proposition 2.2(ii) along with the reflexivity of $W^{1,\mathcal {H}}_0(\Omega )$, there exist a subsequence, still denoted by $\{u_n\}_{n\in {\mathbb {N}}}$, and an element $u_\lambda \in W^{1,\mathcal {H}}_0(\Omega )$ such that

$$\begin{aligned} u_n\rightharpoonup u_\lambda \quad \text {in }W^{1,\mathcal {H}}_0(\Omega ),\quad u_n\rightarrow u_\lambda \quad \text {in }L^{s}(\Omega ) \quad \text {and}\quad u_n\rightarrow u_\lambda \quad \text {a.e. in }\Omega \end{aligned}$$

(3.10)

for any $s\in [1,p^*)$. By the coercivity given in Lemma 3.1(i), we can assume that there exist $E_1,\;E_2\ge 0$ such that

$$\begin{aligned} \lim _{n\rightarrow \infty }\Vert u_n\Vert _p^p=E_1 \quad \text {and}\quad \lim _{n \rightarrow \infty }\Vert u_n\Vert _{q,a}^q=E_2. \end{aligned}$$

(3.11)

We get the following technical results.

Lemma 3.6

Let hypotheses ($\hbox {h}_1$)-($\hbox {h}_2$) be satisfied, let $\lambda \in (0,\Lambda ^*)$ and let $\{u_n\}_{n\in {\mathbb {N}}}\subset \mathcal {N}_\lambda ^+$ be a sequence satisfying (3.8)–(3.9). Then $u_\lambda \ne 0$.

Proof

Let us assume by contradiction that $u_\lambda =0$. Then $\psi '_{u_n}(1)=0$ implies

$$\begin{aligned} \left[ a_0+b_0\phi _{{\mathcal {H}}}^{\theta -1}(\nabla u_n)\right] (\Vert u_n\Vert _p^p+ \Vert u_n\Vert _{q,a}^q)-\lambda \int _\Omega |u_n|^{1-\gamma }\,\mathrm {d}x - \int _\Omega |u_n|^{p^*}\,\mathrm {d}x=0. \end{aligned}$$

Using (3.10), (3.11) and letting $n\rightarrow \infty$, we get

$$\begin{aligned} \left[ a_0+b_0\left( \frac{E_1}{p}+\frac{E_2}{q}\right) ^{\theta -1}\right] (E_1+ E_2) - d^{p^*}=0, \end{aligned}$$

(3.12)

where we set

$$\begin{aligned} \lim _{n\rightarrow \infty }\int _\Omega |u_n|^{p^*}\,\mathrm {d}x=:d^{p^*}\ge 0. \end{aligned}$$

Moreover by (3.8) we have

$$\begin{aligned} \lim _{n\rightarrow \infty }J_\lambda (u_n)=\Theta _\lambda ^+<0, \end{aligned}$$

which implies that

$$\begin{aligned} \left[ a_0\left( \frac{E_1}{p}+\frac{E_2}{q}\right) + b_0\left( \frac{E_1}{p}+\frac{E_2}{q}\right) ^{\theta }\right] -\frac{d^{p^*}}{p^*}<0. \end{aligned}$$

(3.13)

Recall that $E_1$, $E_2\ge 0$. Then, taking the value of $d^{p^*}$ from (3.12) into (3.13), we derive that

$$\begin{aligned} \left[ a_0\left( \frac{E_1}{p}+\frac{E_2}{q}\right) + b_0\left( \frac{E_1}{p}+\frac{E_2}{q}\right) ^{\theta }\right] -\left[ a_0+b_0\left( \frac{E_1}{p}+\frac{E_2}{q}\right) ^{\theta -1}\right] \frac{E_1+E_2}{p^*}<0. \end{aligned}$$

This implies

$$\begin{aligned} a_0 \left[ \frac{E_1}{p}+\frac{E_2}{q}-\frac{E_1+E_2}{p^*}\right] + b_0\left[ \frac{1}{\theta }\left( \frac{E_1}{p}+\frac{E_2}{q}\right) ^\theta -\left( \frac{E_1}{p}+\frac{E_2}{q}\right) ^{\theta -1}\frac{E_1+E_2}{p^*}\right] <0 \end{aligned}$$

and so

$$\begin{aligned}&a_0 \left[ E_1\left( \frac{1}{p}-\frac{1}{p^*}\right) +E_2\left( \frac{1}{q}-\frac{1}{p^*}\right) \right] + b_0\left( \frac{E_1}{p}+\frac{E_2}{q}\right) ^{\theta -1}\\&\quad \left[ E_1\left( \frac{1}{p\theta }-\frac{1}{p^*}\right) + E_2\left( \frac{1}{q\theta }-\frac{1}{p^*}\right) \right] <0, \end{aligned}$$

which is a contradiction because of $p<q\le q\theta <p^*$. $\square$

Lemma 3.7

Let hypotheses ($\hbox {h}_1$)–($\hbox {h}_2$) be satisfied, let $\lambda \in (0,\Lambda ^*)$ and let $\{u_n\}_{n\in {\mathbb {N}}}\subset \mathcal {N}_\lambda ^+$ be a sequence satisfying (3.8)–(3.9). Then $\liminf \limits _{n\rightarrow \infty }\psi ''_{u_n}(1)>0$, that is,

$$\begin{aligned}&\liminf _{n\rightarrow \infty }\bigg \{ \left[ a_0+b_0\phi _{{\mathcal {H}}}^{\theta -1}(\nabla u_n)\right] \left[ (p-1+\gamma )\Vert \nabla u_n\Vert _p^p+(q-1+\gamma )\Vert \nabla u_n\Vert _{q,a}^q \right] \\&\qquad \qquad +b_0(\theta -1)\phi _{{\mathcal {H}}}^{\theta -2}(\nabla u_n)(\Vert \nabla u_n\Vert _p^p + \Vert \nabla u_n\Vert ^{q}_{q,a})^2- { (p^*-1+\gamma )}\int _\Omega |u_n|^{p^*}\,\mathrm {d}x\bigg \}>0. \end{aligned}$$

Proof

Since $\{u_n\}_{n\in {\mathbb {N}}}\subset {\mathcal {N}}_\lambda ^+$, we have $\psi _{u_n}'(1) =0$ and $\psi _{u_n}''(1)>0$, that is,

$$\begin{aligned}&\left[ a_0+b_0\phi _{{\mathcal {H}}}^{\theta -1}(\nabla u_n)\right] \left[ (p-1+\gamma )\Vert \nabla u_n\Vert _p^p+(q-1+\gamma )\Vert \nabla u_n\Vert _{q,a}^q \right] \\&\quad +b_0(\theta -1)\phi _{{\mathcal {H}}}^{\theta -2}(\nabla u_n)(\Vert \nabla u_n\Vert _p^p + \Vert \nabla u_n\Vert ^{q}_{q,a})^2 -(p^*-1+\gamma )\int _\Omega |u_n|^{p^*}\,\mathrm {d}x>0 \end{aligned}$$

and

$$\begin{aligned} \begin{aligned}&\left[ a_0+b_0\phi _{{\mathcal {H}}}^{\theta -1}(\nabla u_n)\right] \left[ (p-p^*)\Vert \nabla u_n\Vert _p^p+(q-p^*)\Vert \nabla u_n\Vert _{q,a}^q \right] \\&\quad +b_0(\theta -1)\phi _{{\mathcal {H}}}^{\theta -2}(\nabla u_n)(\Vert \nabla u_n\Vert _p^p + \Vert \nabla u_n\Vert ^{q}_{q,a})^2 +\lambda (p^*-1+\gamma )\int _\Omega |u_n|^{1-\gamma }\,\mathrm {d}x>0. \end{aligned} \end{aligned}$$

(3.14)

Thus, in order to prove the lemma, it is enough to show that

$$\begin{aligned}&\liminf _{n\rightarrow \infty }\bigg \{\left[ a_0+b_0\phi _{{\mathcal {H}}}^{\theta -1}(\nabla u_n)\right] \left[ (p-p^*)\Vert \nabla u_n\Vert _p^p+(q-p^*)\Vert \nabla u_n\Vert _{q,a}^q \right] \\&\qquad \qquad +b_0(\theta -1)\phi _{{\mathcal {H}}}^{\theta -2}(\nabla u_n)(\Vert \nabla u_n\Vert _p^p + \Vert \nabla u_n\Vert ^{q}_{q,a})^2 \\&\qquad \qquad +\lambda (p^*-1+\gamma )\int _\Omega |u_n|^{1-\gamma }\,\mathrm {d}x\bigg \}>0. \end{aligned}$$

By contradicting (3.14), let us assume that

$$\begin{aligned} \begin{aligned}&\liminf _{n\rightarrow \infty }\bigg \{ \left[ a_0+b_0\phi _{{\mathcal {H}}}^{\theta -1}(\nabla u_n)\right] \left[ (p-p^*)\Vert \nabla u_n\Vert _p^p+(q-p^*)\Vert \nabla u_n\Vert _{q,a}^q \right] \\&\quad +b_0(\theta -1)\phi _{{\mathcal {H}}}^{\theta -2}(\nabla u_n)(\Vert \nabla u_n\Vert _p^p + \Vert \nabla u_n\Vert ^{q}_{q,a})^2 +\lambda (p^*-1+\gamma )\int _\Omega |u_n|^{1-\gamma }\,\mathrm {d}x\bigg \}=0. \end{aligned} \end{aligned}$$

(3.15)

By Lebesgue dominated convergence theorem, we obtain

$$\begin{aligned} \lim _{n\rightarrow \infty }\int _\Omega |u_n|^{1-\gamma }\,\mathrm {d}x = \int _\Omega |u_\lambda |^{1-\gamma }\mathrm {d}x. \end{aligned}$$

(3.16)

Using (3.16) in (3.15), we get

$$\begin{aligned}&\liminf _{n\rightarrow \infty }\bigg \{ \left[ a_0+b_0\phi _{{\mathcal {H}}}^{\theta -1}(\nabla u_n)\right] \left[ (p-p^*)\Vert \nabla u_n\Vert _p^p+(q-p^*)\Vert \nabla u_n\Vert _{q,a}^q \right] \\&\quad +b_0(\theta -1)\phi _{{\mathcal {H}}}^{\theta -2}(\nabla u_n)(\Vert \nabla u_n\Vert _p^p + \Vert \nabla u_n\Vert ^{q}_{q,a})^2\bigg \}\\&=-\lambda (p^*-1+\gamma )\int _\Omega |u_\lambda |^{1-\gamma }\,\mathrm {d}x, \end{aligned}$$

which yields, by applying (3.11),

$$\begin{aligned} \begin{aligned} -\lambda \int _\Omega |u_\lambda |^{1-\gamma }\,\mathrm {d}x&= \left[ a_0+b_0\left( \frac{E_1}{p}+\frac{E_2}{q} \right) ^{\theta -1}\right] \frac{\left[ (p-p^*)E_1+(q-p^*)E_2 \right] }{(p^*-1+\gamma )}\\&\quad + \frac{b_0(\theta -1)}{(p^*-1+\gamma )}\left( \frac{E_1}{p}+\frac{E_2}{q}\right) ^{\theta -2}(E_1 + E_2)^2. \end{aligned} \end{aligned}$$

(3.17)

From this, due to $p<q< p^*$, we have

$$\begin{aligned} \begin{aligned} -\lambda \int _\Omega |u_\lambda |^{1-\gamma }\,\mathrm {d}x&\le b_0\left( \frac{E_1}{p}+\frac{E_2}{q}\right) ^{\theta -1} \left[ \frac{(q-p^*)(E_1+E_2)}{(p^*-1+\gamma )} + \frac{b_0(\theta -1)q(E_1+E_2)}{(p^*+\gamma -1)}\right] \\&= \frac{b_0(q \theta -p^*)(E_1+E_2)}{(p^*+\gamma -1)} \left( \frac{E_1}{p}+\frac{E_2}{q}\right) ^{\theta -1}. \end{aligned} \end{aligned}$$

(3.18)

Considering $\psi _{u_n}'(1)=0$ and (3.16), we have

$$\begin{aligned} \lim _{n\rightarrow \infty }\int _\Omega |u_n|^{p^*}\,\mathrm {d}x= \left[ a_0+b_0\left( \frac{E_1}{p}+\frac{E_2}{q}\right) ^{\theta -1}\right] \left[ E_1+E_2 \right] -\lambda \int _{\Omega }|u_\lambda |^{1-\gamma }\,\mathrm {d}x. \end{aligned}$$

From this and (3.17), we obtain

$$\begin{aligned} \begin{aligned}&\lim _{n\rightarrow \infty }\int _\Omega |u_n|^{p^*}\,\mathrm {d}x\\&= \left[ a_0+b_0\left( \frac{E_1}{p}+\frac{E_2}{q}\right) ^{\theta -1}\right] \left[ \left( \frac{p+\gamma -1}{p^*+\gamma -1}\right) E_1 +\left( \frac{q+\gamma -1}{p^*+\gamma -1}\right) E_2\right] \\&\qquad + \frac{b_0(\theta -1)}{p^*-1+\gamma }\left( \frac{E_1}{p}+\frac{E_2}{q}\right) ^{\theta -2}(E_1+E_2)^2\\&\ge \frac{b_0 (p+\gamma -1)}{p^*+\gamma -1} \left( \frac{E_1}{p}+\frac{E_2}{q}\right) ^{\theta -1} (E_1 +E_2) + \frac{b_0 (p\theta -p)}{p^*-1+\gamma }\left( \frac{E_1}{p}+\frac{E_2}{q}\right) ^{\theta -1}(E_1+E_2 ) \\&= \frac{b_0 (p\theta +\gamma -1)}{p^*+\gamma -1} \left( \frac{E_1}{p}+\frac{E_2}{q}\right) ^{\theta -1} (E_1 +E_2)\\&\ge \frac{b_0 (p\theta +\gamma -1)}{p^{\theta -1}(p^*+\gamma -1)} E_1^{\theta }. \end{aligned} \end{aligned}$$

(3.19)

For any fixed $w\in W^{1,\mathcal {H}}_0(\Omega )\setminus \{0\}$, we know that there exists a unique $t_{\max }>0$ such that $\sigma '_w(t_{\max })=0$. From this and (3.6), we conclude that

$$\begin{aligned} t_{\max }\ge \left( \frac{b_0(p \theta +\gamma -1)\Vert \nabla w\Vert _p^{p\theta }}{p^{\theta -1}(p^*-1+\gamma )\int _\Omega |w|^{p^*}\,\mathrm {d}x}\right) ^{\frac{1}{p^*-p\theta }}:= t_{00}. \end{aligned}$$

(3.20)

It is easy to verify that $t_{\max }\ge t_{00}\ge t^w_0$ as defined in (3.7) and from the proof of Lemma 3.2, we know that $\Lambda _2>0$ is chosen in such a way that

$$\begin{aligned} \frac{b_0(p^*-q \theta )}{p^{\theta -1}(p^*+\gamma -1)}(t_0^w)^{p\theta +\gamma -1}\Vert \nabla w\Vert _p^{p\theta } \ge \Lambda _2 \int _\Omega |w|^{1-\gamma }\mathrm {d}x. \end{aligned}$$

We define

$$\begin{aligned} S(w):= & {} \frac{b_0(p^*- q \theta )}{p^{\theta -1}(p^*+\gamma -1)}(t_{00})^{p\theta +\gamma -1}\Vert \nabla w\Vert _p^{p\theta }- \Lambda _2 \int _\Omega |w|^{1-\gamma }\mathrm {d}x \ge 0\nonumber \\&\quad \text {for all }w\in W^{1,\mathcal {H}}_0(\Omega ), \end{aligned}$$

(3.21)

with $t_{00}$ given in (3.20). Taking $w= u_n$ in (3.21) and then passing to the limit as $n\rightarrow \infty$ we get

$$\begin{aligned} \lim _{n\rightarrow \infty } S(u_n)\ge 0. \end{aligned}$$

On the other hand, by Lemma 3.6 and (3.11), we have that at least one of $E_1$ and $E_2$ is not zero. Let us assume, without any loss of generality, that $E_1>0$, $E_2 \ge 0$. Then by (3.18), (3.19), (3.20) along with $q\theta < p^*$ and $\lambda \in (0, \Lambda _2)$, we obtain

$$\begin{aligned} \lim _{n\rightarrow \infty } S(u_n)&\le \frac{ \frac{b_0(p^*- q \theta )}{p^{\theta -1}(p^*+\gamma -1)} \left( \frac{b_0(p \theta +\gamma -1) E_1^\theta }{p^{\theta -1}(p^*-1+\gamma )}\right) ^{\frac{(p\theta -1+\gamma )}{p^*-p\theta }} E_1^\theta }{\left( \frac{b_0 (p\theta +\gamma -1)}{p^{\theta -1}(p^*+\gamma -1)} E_1^{\theta }\right) ^{\frac{p\theta +\gamma -1}{p^*-p\theta }}} \\ {}&\quad + \frac{\Lambda _2}{\lambda }\frac{b_0(q \theta -p^*)(E_1+E_2)}{(p^*+\gamma -1)} \left( \frac{E_1}{p}+\frac{E_2}{q}\right) ^{\theta -1}\\&< \frac{b_0(p^*- q \theta )}{p^{\theta -1}(p^*+\gamma -1)} E_1^\theta + \frac{b_0(q \theta -p^*)E_1^\theta }{p^{\theta -1}(p^*+\gamma -1)} =0. \end{aligned}$$

This proves the assertion of the lemma. $\square$

Let $h\in W^{1,\mathcal {H}}_0(\Omega )$ be nonnegative. From Lemma 3.5 there exists a sequence of maps $\{\zeta _n\}_{n\in {\mathbb {N}}}$ such that $\zeta _n(0) =1$ and $\zeta _n(th)(u_n+th)\in {\mathcal {N}}_\lambda ^+$ for sufficiently small $t>0$ and for each $n\in {\mathbb {N}}$. From this and $u_n\in {\mathcal {N}}_\lambda$, we have the equations

$$\begin{aligned} \left[ a_0 +b_0\phi _\mathcal {H}^{\theta -1}(\nabla u_n)\right] \left( \Vert \nabla u_n\Vert _p^p+ \Vert \nabla u_n\Vert _{q,a}^q\right) - \lambda \int _\Omega |u_n|^{1-\gamma }\,\mathrm {d}x - \int _\Omega |u_n|^{p^*}\,\mathrm {d}x=0 \end{aligned}$$

(3.22)

and

$$\begin{aligned} \begin{aligned}&\left[ a_0 +b_0\phi _\mathcal {H}^{\theta -1}(\zeta _n(th)\nabla w_n)\right] \left( \zeta _n^{p}(th)\Vert \nabla w_n\Vert _p^p+\zeta _n^{q}(th) \Vert \nabla w_n\Vert _{q,a}^q\right) \\&\quad - \lambda \zeta _n^{1-\gamma }(th)\int _\Omega |w_n|^{1-\gamma }\,\mathrm {d}x - \zeta _n^{p^*}(th)\int _\Omega |w_n|^{p^*}\,\mathrm {d}x =0 \end{aligned} \end{aligned}$$

(3.23)

where $w_n = u_n+th$.

Lemma 3.8

Let hypotheses ($\hbox {h}_1$)-($\hbox {h}_2$) be satisfied, let $\lambda \in (0,\Lambda ^*)$ and let $\{u_n\}_{n\in {\mathbb {N}}}\subset \mathcal {N}_\lambda ^+$ be a sequence satisfying (3.8)–(3.9). For any nonnegative function $h\in W^{1,\mathcal {H}}_0(\Omega )$, the sequence $\{\langle \zeta '_n(0),h \rangle \}_{n\in {\mathbb {N}}}$ is uniformly bounded.

Proof

Subtracting (3.22) from (3.23), we get

$$\begin{aligned} \begin{aligned}&(a_0 + b_0 \phi _{{\mathcal {H}}}^{\theta -1}(\nabla u_n))\left[ (\Vert \nabla w_n\Vert _p^p-\Vert \nabla u_n\Vert _p^p) + (\Vert \nabla w_n\Vert _{q,a}^q-\Vert \nabla u_n\Vert _{q,a}^q) \right. \\&\quad \left. +\, (\zeta _n^p(th)-1)\Vert \nabla w_n\Vert _p^p+ (\zeta _n^q(th)-1)\Vert \nabla w_n\Vert _{q,a}^q\right] \\&\quad +\, b_0 \left[ \phi _{{\mathcal {H}}}^{\theta -1}(\zeta _n(th)\nabla w_n) - \phi _{{\mathcal {H}}}^{\theta -1}(\nabla u_n)\right] \left( \zeta _n^p(th)\Vert \nabla w_n\Vert _p^p+ \zeta _n^q(th)\Vert \nabla w_n\Vert _{q,a}^q\right) \\&\quad -\lambda \left( \zeta _n^{1-\gamma }(th)-1\right) \int _\Omega |w_n|^{1-\gamma }\,\mathrm {d}x -\lambda \int _\Omega \left( |w_n|^{1-\gamma }-|u_n|^{1-\gamma }\right) \,\mathrm {d}x\\&\quad - \left( \zeta _n^{p^*}(th)-1\right) \int _\Omega |w_n|^{p^*}\,\mathrm {d}x - \int _\Omega \left( |w_n|^{p^*}-|u_n|^{p^*}\right) \,\mathrm {d}x =0. \end{aligned} \end{aligned}$$

(3.24)

For notational convenience, we set

$$\begin{aligned} \langle u_n,h \rangle _p = \int _\Omega |\nabla u_n|^{p-2}\nabla u_n\cdot \nabla h\,\mathrm {d}x \quad \text {and}\quad \langle u_n,h \rangle _{q,a} = \int _\Omega a(x)|\nabla u_n|^{q-2}\nabla u_n\cdot \nabla h\,\mathrm {d}x. \end{aligned}$$

We have the following limits

$$\begin{aligned} \begin{aligned} \lim _{t\rightarrow 0} \frac{\phi _{{\mathcal {H}}}^{\theta -1}(\zeta _n(th)\nabla w_n)- \phi _{{\mathcal {H}}}^{\theta -1}(\nabla u_n)}{t}&= (\zeta _n'(0),h)(\theta -1)\phi _{{\mathcal {H}}}^{\theta -2}(\nabla u_n)(\Vert \nabla u_n\Vert _p^p+ \Vert \nabla u_n\Vert _{q,a}^q)\\&\quad + (\theta -1)\phi _{{\mathcal {H}}}^{\theta -2}(\nabla u_n)(\langle u_n,h \rangle _p+\langle u_n,h \rangle _{q,a}),\\ \lim _{t\rightarrow 0} \frac{\Vert \nabla w_n\Vert _p^p-\Vert \nabla u_n\Vert _p^p}{t}&= p\langle u_n,h \rangle _p,\\ \lim _{t\rightarrow 0} \frac{\Vert \nabla w_n\Vert _{q,a}^q-\Vert \nabla u_n\Vert _{q,a}^q}{t}&= q\langle u_n,h \rangle _{q,a} ,\\ \lim _{t\rightarrow 0}\left( |w_n|^{p^*}-|u_n|^{p^*}\right) \,\mathrm {d}x&= p^*\int _\Omega |u_n|^{p^*-2}u_n h\,\mathrm {d}x,\\ \lim _{t\rightarrow 0} \frac{\zeta ^s(th)-1}{t}&= s \langle \zeta _n'(0),h \rangle \quad \text {for any } s >1. \end{aligned} \end{aligned}$$

(3.25)

Taking into account

$$\begin{aligned} \int _\Omega \left( |w_n|^{1-\gamma }-|u_n|^{1-\gamma }\right) \,\mathrm {d}x\ge 0 \end{aligned}$$

since h is nonnegative, dividing both sides of (3.24) by $t>0$ and then passing the limit as $t\rightarrow 0^+$, we obtain

$$\begin{aligned} 0&\le (a_0 + b_0 \phi _{{\mathcal {H}}}^{\theta -1}(\nabla u_n))\left( p\int _\Omega |\nabla u_n|^{p-2}\nabla u_n\nabla h\,\mathrm {d}x + q\int _\Omega a(x)|\nabla u_n|^{q-2}\nabla u_n\nabla h\,\mathrm {d}x \right. \\&\qquad + p\ \langle \zeta '_n(0),h \rangle \Vert \nabla u_n\Vert _p^p+ q \langle \zeta '_n(0),h \rangle \Vert \nabla u_n\Vert _{q,a}^q \bigg ) \\&\qquad +b_0 (\theta -1)\phi _{{\mathcal {H}}}^{\theta -2}(\nabla u_n)\langle \zeta '_n(0),h \rangle \left( \Vert \nabla u_n\Vert _p^p+ \Vert \nabla u_n\Vert _{q,a}^q \right) ^2\\&\qquad -\lambda (1-\gamma )\langle \zeta '_n(0),h \rangle \int _\Omega |u_n|^{1-\gamma }\,\mathrm {d}x -p^*\langle \zeta '_n(0),h \rangle \int _\Omega |u_n|^{p^*}\mathrm {d}x-p^*\int _\Omega |u_n|^{p^*-2}u_n h\,\mathrm {d}x. \end{aligned}$$

This implies

$$\begin{aligned} 0&\le \langle \zeta _n'(0),h \rangle \bigg [(a_0 + b_0 \phi _{{\mathcal {H}}}^{\theta -1}(\nabla u_n)) \left[ p\Vert \nabla u_n\Vert _p^p+ q\Vert \nabla u_n\Vert _{q,a}^q \right] \\&\quad + b_0 (\theta -1)\phi _{{\mathcal {H}}}^{\theta -2}(\nabla u_n) \left( \Vert \nabla u_n\Vert _p^p+ \Vert \nabla u_n\Vert _{q,a}^q \right) ^2 \\ {}&\quad -\,\lambda (1-\gamma ) \int _\Omega |u_n|^{1-\gamma }\,\mathrm {d}x-p^*\int _\Omega |u_n|^{p^*}\mathrm {d}x \bigg ]+ (a_0 + b_0 \phi _{{\mathcal {H}}}^{\theta -1}(\nabla u_n)) \\&\quad \left( p\int _\Omega |\nabla u_n|^{p-2}\nabla u_n\cdot \nabla h\,\mathrm {d}x + q\int _\Omega a(x)|\nabla u_n|^{q-2}\nabla u_n\cdot \nabla h\,\mathrm {d}x\right) \\&\quad -p^*\int _\Omega |u_n|^{p^*-2}u_n h\,\mathrm {d}x. \end{aligned}$$

Therefore, using the fact that $u_n \in \mathcal {N}_\lambda$, we have

$$\begin{aligned} 0&\le \langle \zeta _n'(0),h \rangle \bigg \{(a_0 + b_0 \phi _{{\mathcal {H}}}^{\theta -1}(\nabla u_n)) \left[ (p+\gamma -1)\Vert \nabla u_n\Vert _p^p+ (q+\gamma -1)\Vert \nabla u_n\Vert _{q,a}^q \right] \\&\quad +b_0(\theta -1)\phi _{{\mathcal {H}}}^{\theta -2}(\nabla u_n)(\Vert \nabla u_n\Vert _p^p + \Vert \nabla u_n\Vert ^{q}_{q,a})^2 -(p^*+\gamma -1)\int _\Omega |u_n|^{p^*}\mathrm {d}x \bigg \}\\&\quad + \bigg [(a_0 + b_0 \phi _{{\mathcal {H}}}^{\theta -1}(\nabla u_n)\left( p\int _\Omega |\nabla u_n|^{p-2}\nabla u_n\cdot \nabla h\,\mathrm {d}x + q\int _\Omega a(x)|\nabla u_n|^{q-2}\nabla u_n\cdot \nabla h\,\mathrm {d}x\right) \\&\quad -p^*\int _\Omega |u_n|^{p^*-2}u_n h\,\mathrm {d}x\bigg ]. \end{aligned}$$

Now using Lemma 3.7 and taking into account the boundedness of $\{u_n\}_{n\in {\mathbb {N}}}$ in $W^{1,\mathcal {H}}_0(\Omega )$, we infer that $\{\langle \zeta _n'(0),h \rangle \}_{n\in {\mathbb {N}}}$ is bounded below for any nonnegative $h\in W^{1,\mathcal {H}}_0(\Omega )$.

It remains to show that $\{\langle \zeta _n'(0),h \rangle \}_{n\in {\mathbb {N}}}$ is bounded above for any nonnegative $h\in W^{1,\mathcal {H}}_0(\Omega )$. Assume by contradiction that $\limsup _{n\rightarrow \infty } \langle \zeta _n'(0),h \rangle =\infty$. Thus, without loss of generality, we can consider $\zeta _n(th)>\zeta _n(0)=1$ for $n\in {\mathbb {N}}$ large enough . It is easy to see that

$$\begin{aligned} |\zeta _n(th)-1|\Vert u_n\Vert + \zeta _n(th)\Vert th\Vert \ge \Vert (\zeta _n(th)-1)u_n + th\zeta _n(th)\Vert = \Vert \zeta _n(th)w_n-u_n\Vert . \end{aligned}$$

Applying this in (3.9) with $u= \zeta _n(th)w_n$, we get

$$\begin{aligned}&|\zeta _n(th)-1|\frac{\Vert u_n\Vert }{n} + \zeta _n(th)\frac{\Vert th\Vert }{n}\\&\ge J_\lambda (u_n) - J_\lambda (\zeta _n(th)w_n)\\&= a_0\left[ \phi _{{\mathcal {H}}}(\nabla u_n)- \phi _{{\mathcal {H}}}(\zeta _n(th)\nabla w_n)\right] +\frac{b_0}{\theta }\left[ \phi _{{\mathcal {H}}}^\theta (\nabla u_n)- \phi _{{\mathcal {H}}}^\theta (\zeta _n(th)\nabla w_n)\right] \\&\quad -\frac{\lambda }{1-\gamma }\int _\Omega \left[ |u_n|^{1-\gamma }-|\zeta _n(th)w_n|^{1-\gamma }\right] \,\mathrm {d}x-\frac{1}{p^*}\int _\Omega \left[ |u_n|^{p^*}-|\zeta _n(th)w_n|^{p^*}\right] \,\mathrm {d}x. \end{aligned}$$

Using (3.22) and (3.23) in the inequality above, we obtain

$$\begin{aligned}&|\zeta _n(th)-1|\frac{\Vert u_n\Vert }{n}+ \zeta _n(th)\frac{\Vert th\Vert }{n} \\&= a_0\left[ \phi _{{\mathcal {H}}}(\nabla u_n)- \phi _{\mathcal H}(\zeta _n(th)\nabla w_n) -\frac{1}{1-\gamma }\right. \\ {}&\quad \left. \left( \Vert \nabla u_n\Vert _p^p+\Vert \nabla u_n\Vert _{q,a}^q-\zeta _n^p(th)\Vert \nabla w_n\Vert _p^p-\zeta _n^q(th)\Vert \nabla w_n\Vert _{q,a}^q \right) \right] \\&\quad +b_0\bigg [\frac{\phi _{{\mathcal {H}}}^\theta (\nabla u_n)- \phi _{{\mathcal {H}}}^\theta (\zeta _n(th)\nabla w_n)}{\theta } -\frac{\phi _{{\mathcal {H}}}^{\theta -1}(\nabla u_n)}{1-\gamma }\left( \Vert \nabla u_n\Vert _p^p+\Vert \nabla u_n\Vert _{q,a}^q\right) \\&\quad + \frac{\phi _{\mathcal H}^{\theta -1}(\zeta _n(th)\nabla w_n)}{1-\gamma }\left( \zeta _n^p(th)\Vert \nabla w_n\Vert _p^p+\zeta _n^q(th)\Vert \nabla w_n\Vert _{q,a}^q \right) \bigg ]\\&\quad - \left( \frac{1}{1-\gamma }-\frac{1}{p^*}\right) \int _\Omega \left[ |\zeta _n(th)w_n|^{p^*}-|u_n|^{p^*}\right] \,\mathrm {d}x. \end{aligned}$$

Now dividing the above inequality by $t>0$, passing to the limit as $t\rightarrow 0^+$ and using (3.25), we have

$$\begin{aligned} \frac{\Vert h\Vert }{n}&\ge a_0\bigg [\langle u_n,h \rangle _p+\langle u_n,h \rangle _{q,a} -\langle \zeta _n'(0),h \rangle \left( \Vert \nabla u_n\Vert _p^p+\Vert \nabla u_n\Vert _{q,a}^q\right) \\&\quad +\frac{1}{1-\gamma }\bigg \{\langle \zeta _n'(0),h \rangle \left( p\Vert \nabla u_n\Vert _p^p+q\Vert \nabla u_n\Vert _{q,a}^q\right) + p\langle u_n,h \rangle _p+q\langle u_n,h \rangle _{q,a}\bigg \}\bigg ]\\&\quad + b_0\bigg [ \phi _{{\mathcal {H}}}^{\theta -1}(\nabla u_n)\langle \zeta _n'(0),h \rangle \left( p\Vert \nabla u_n\Vert _p^p+q\Vert \nabla u_n\Vert _{q,a}^q\right) \\&\quad +\frac{1}{1-\gamma }\bigg \{ \langle \zeta _n'(0),h \rangle (\theta -1)\phi _{{\mathcal {H}}}^{\theta -2}(\nabla u_n)(\Vert \nabla u_n\Vert _p^p+ \Vert \nabla u_n\Vert _{q,a}^q)^2\\&\quad + \phi _{{\mathcal {H}}}^{\theta -1}(\nabla u_n)\langle \zeta _n'(0),h \rangle \left( p\Vert \nabla u_n\Vert _p^p+q\Vert \nabla u_n\Vert _{q,a}^q\right) + \phi _{{\mathcal {H}}}^{\theta -1}(\nabla u_n)\\&\quad \left( p\langle u_n,h \rangle _p+q\langle u_n,h \rangle _{q,a}\right) \bigg \}\bigg ]\\&\quad -\left( \frac{p^*-1+\gamma }{1-\gamma }\right) \left[ \langle \zeta _n'(0),h \rangle \int _\Omega |u_n|^{p^*}\,\mathrm {d}x+ \int _\Omega |u_n|^{p^*-2}u_nh\,\mathrm {d}x\right] \\&=\frac{ \langle \zeta _n'(0),h \rangle }{1-\gamma }\bigg [ \left( a_0+\phi _{{\mathcal {H}}}^{\theta -1}(\nabla u_n)\right) \bigg \{(p-1+\gamma )\Vert \nabla u_n\Vert _p^p+(q-1+\gamma ) \Vert \nabla u_n\Vert _{q,a}^q\bigg \}\\&\quad + b_0(\theta -1)\phi _{{\mathcal {H}}}^{\theta -2}(\nabla u_n)(\Vert \nabla u_n\Vert _p^p+ \Vert \nabla u_n\Vert _{q,a}^q)^2 \\ {}&\quad - (p^*-1+\gamma )\int _\Omega |u_n|^{p^*}\,\mathrm {d}x -\frac{(1-\gamma )\Vert u_n\Vert }{n}\bigg ]\\&\quad + \frac{a_0}{1-\gamma } \bigg [(p-\gamma +1)\langle u_n,h \rangle _p+(q-\gamma +1)\langle u_n,h \rangle _{q,a}\bigg ] \\ {}&\quad +\frac{b_0\phi _{{\mathcal {H}}}^{\theta -1}(\nabla u_n)}{1-\gamma }\bigg [p\langle u_n,h \rangle _p+q\langle u_n,h \rangle _{q,a}\bigg ]\\&\quad -\left( \frac{p^*-1+\gamma }{1-\gamma }\right) \int _\Omega |u_n|^{p^*-2}u_nh\,\mathrm {d}x, \end{aligned}$$

which gives a contradiction if we take the limits $n\rightarrow \infty$ on both sides, considering $\limsup _{n\rightarrow \infty }\langle \zeta _n'(0),h \rangle =\infty$, since by Lemma 3.7 and the boundedness of $\{u_n\}_{n\in {\mathbb {N}}}$, there exists some $M_1>0$ such that

$$\begin{aligned}&\bigg [ \left( a_0+\phi _{{\mathcal {H}}}^{\theta -1}(\nabla u_n)\right) \bigg \{(p-1+\gamma )\Vert \nabla u_n\Vert _p^p+(q-1+\gamma ) \Vert \nabla u_n\Vert _{q,a}^q\bigg \}\\&\quad + b_0(\theta -1)\phi _{{\mathcal {H}}}^{\theta -2}(\nabla u_n)(\Vert \nabla u_n\Vert _p^p+ \Vert \nabla u_n\Vert _{q,a}^q)^2 \\&\quad - (p^*-1+\gamma )\int _\Omega |u_n|^{p^*}\,\mathrm {d}x -\frac{(1-\gamma )\Vert u_n\Vert }{n}\bigg ] > M_1 \end{aligned}$$

for $n\in {\mathbb {N}}$ large enough. Thus $\{\langle \zeta _n'(0),h \rangle \}_{n\in {\mathbb {N}}}$ must be bounded above. $\square$

Since $J_\lambda (u_n)=J_\lambda (|u_n|)$, without loss of generality, we may assume that $u_n\ge 0$ a. e. in $\Omega$ and so, $u_\lambda \ge 0$ a. e. in $\Omega$. With this assumption, we state our next result.

Lemma 3.9

Let hypotheses ($\hbox {h}_1$)-($\hbox {h}_2$) be satisfied, let $\lambda \in (0,\Lambda ^*)$ and let $\{u_n\}_{n\in {\mathbb {N}}}\subset \mathcal {N}_\lambda ^+$ be a sequence satisfying (3.8)–(3.9). For any $h\in W^{1,\mathcal {H}}_0(\Omega )$ and $n\in {\mathbb {N}}$, $u_n^{-\gamma }h \in L^1(\Omega )$ and as $n\rightarrow \infty$

$$\begin{aligned}&(a_0+b_0\phi _{{\mathcal {H}}}^{\theta -1}(\nabla u_n)) \left[ \int _\Omega |\nabla u_n|^{p-2}\nabla u_n\cdot \nabla h\,\mathrm {d}x + \int _\Omega a(x)|\nabla u_n|^{q-2}\nabla u_n\cdot \nabla h\,\mathrm {d}x\right] \nonumber \\&\quad -\lambda \int _\Omega u_n^{-\gamma }h\,\mathrm {d}x - \int _\Omega u_n^{p^*-1}h\,\mathrm {d}x= o_n(1). \end{aligned}$$

(3.26)

Proof

Let $h\in W^{1,\mathcal {H}}_0(\Omega )$ be nonnegative and recall the following estimate from the proof of Lemma 3.8

$$\begin{aligned}&|\zeta _n(th)-1|\frac{\Vert u_n\Vert }{n}+ \zeta _n(th)\frac{\Vert th\Vert }{n} \\&\ge a_0\left[ \phi _{{\mathcal {H}}}(\nabla u_n)- \phi _{{\mathcal {H}}}(\zeta _n(th)\nabla w_n)\right] +\frac{b_0}{\theta }\left[ \phi _{{\mathcal {H}}}^\theta (\nabla u_n)- \phi _{{\mathcal {H}}}^\theta (\zeta _n(th)\nabla w_n)\right] \\&\quad -\frac{\lambda }{1-\gamma }\int _\Omega \left[ |u_n|^{1-\gamma }-|\zeta _n(th)w_n|^{1-\gamma }\right] \,\mathrm {d}x-\frac{1}{p^*}\int _\Omega \left[ |u_n|^{p^*}-|\zeta _n(th)w_n|^{p^*}\right] \,\mathrm {d}x\\&= a_0 \left[ (\phi _{{\mathcal {H}}}(\nabla u_n)- \phi _{{\mathcal {H}}}(\nabla w_n))+ (\phi _{{\mathcal {H}}}(\nabla w_n)- \phi _{{\mathcal {H}}}(\zeta _n(th)\nabla w_n)\right) ]\\&\quad +\frac{b_0}{\theta }\left[ (\phi _{{\mathcal {H}}}^\theta (\nabla u_n)-\phi _{{\mathcal {H}}}^\theta (\nabla w_n)) +(\phi _{\mathcal H}^\theta (\nabla w_n)-\phi _{{\mathcal {H}}}^\theta (\zeta _n(th)\nabla w_n))\right] \\&\quad -\frac{\lambda }{1-\gamma }\int _\Omega \left[ |u_n|^{1-\gamma }-|w_n|^{1-\gamma }\right] \,\mathrm {d}x -\frac{\lambda }{1-\gamma }\int _\Omega \left[ |w_n|^{1-\gamma }-|\zeta _n(th)w_n|^{1-\gamma }\right] \,\mathrm {d}x\\&\quad -\frac{1}{p^*}\int _\Omega \left[ |u_n|^{p^*}-|w_n|^{p^*}\right] \,\mathrm {d}x-\frac{1}{p^*}\int _\Omega \left[ |w_n|^{p^*}-|\zeta _n(th)w_n|^{p^*}\right] \,\mathrm {d}x. \end{aligned}$$

Dividing the above equation with $t>0$ and then passing to limit as $t\rightarrow 0^+$, we get

$$\begin{aligned}&|\langle \zeta '_n(0), h \rangle |\frac{\Vert u_n\Vert }{n}+ \frac{\Vert h\Vert }{n}\\&\ge -(a_0+b_0\phi _{{\mathcal {H}}}^{\theta -1}(\nabla u_n)) \left[ \langle u_n,h \rangle _p + \langle u_n,h \rangle _{q,a}+ \langle \zeta '_n(0), h \rangle (\Vert u_n\Vert _p^p+ \Vert u_n\Vert _{q,a}^q)\right] \\&\quad -\frac{\lambda }{1-\gamma }\liminf _{t\rightarrow 0^+}\int _\Omega \frac{\left[ |u_n|^{1-\gamma }-|w_n|^{1-\gamma }\right] }{t}\,\mathrm {d}x + \lambda \langle \zeta '_n(0), h \rangle \int _\Omega |u_n|^{1-\gamma }\,\mathrm {d}x \\&\quad \quad + \langle \zeta '_n(0), h \rangle \int _\Omega |u_n|^{p^*}\,\mathrm {d}x + \int _\Omega u_n^{p^*-1}h\,\mathrm {d}x\\&= - \langle \zeta '_n(0), h \rangle \bigg [(a_0+b_0\phi _{{\mathcal {H}}}^{\theta -1}(\nabla u_n)) \left[ (\Vert u_n\Vert _p^p+ \Vert u_n\Vert _{q,a}^q)\right] \\ {}&\quad -\lambda \int _\Omega |u_n|^{1-\gamma }\,\mathrm {d}x- \int _\Omega |u_n|^{p^*}\,\mathrm {d}x\bigg ]\\&\quad - (a_0+b_0\phi _{{\mathcal {H}}}^{\theta -1}(\nabla u_n)) \left[ \langle u_n,h \rangle _p + \langle u_n,h \rangle _{q,a}\right] \\&\quad -\frac{\lambda }{1-\gamma }\liminf _{t\rightarrow 0^+}\int _\Omega \frac{\left[ |u_n|^{1-\gamma }-|w_n|^{1-\gamma }\right] }{t}\,\mathrm {d}x + \int _\Omega u_n^{p^*-1}h\,\mathrm {d}x\\&= - (a_0+b_0\phi _{{\mathcal {H}}}^{\theta -1}(\nabla u_n)) \left[ \langle u_n,h \rangle _p + \langle u_n,h \rangle _{q,a}\right] \\&\quad -\frac{\lambda }{1-\gamma }\liminf _{t\rightarrow 0^+}\int _\Omega \frac{\left[ |u_n|^{1-\gamma }-|w_n|^{1-\gamma }\right] }{t}\,\mathrm {d}x + \int _\Omega u_n^{p^*-1}h\,\mathrm {d}x, \end{aligned}$$

where we used $u_n\in {\mathcal {N}}_\lambda$ that is $\psi '_{u_n}(1)=0$. This implies

$$\begin{aligned} \begin{aligned}&\frac{\lambda }{1-\gamma }\liminf _{t\rightarrow 0^+}\int _\Omega \frac{\left[ |u_n+th|^{1-\gamma }-|u_n|^{1-\gamma }\right] }{t}\,\mathrm {d}x \\ {}&\quad \le (a_0+b_0\phi _{{\mathcal {H}}}^{\theta -1}(\nabla u_n)) \left[ \langle u_n,h \rangle _p + \langle u_n,h \rangle _{q,a}\right] \\&\qquad - \int _\Omega u_n^{p^*-1}h\,\mathrm {d}x + |\langle \zeta '_n(0), h \rangle |\frac{\Vert u_n\Vert }{n}+ \frac{\Vert h\Vert }{n}. \end{aligned} \end{aligned}$$

(3.27)

Observe that $|u_n+th|^{1-\gamma }-|u_n|^{1-\gamma }\ge 0$, so we can use Fatou’s lemma in (3.27) to obtain

$$\begin{aligned} \lambda \int _\Omega u_n^{-\gamma }h \,\mathrm {d}x&\le (a_0+b_0\phi _{{\mathcal {H}}}^{\theta -1}(\nabla u_n)) \left[ \langle u_n,h \rangle _p + \langle u_n,h \rangle _{q,a}\right] \nonumber \\&\quad - \int _\Omega u_n^{p^*-1}h\,\mathrm {d}x + |\langle \zeta '_n(0), h \rangle |\frac{\Vert u_n\Vert }{n}+ \frac{\Vert h\Vert }{n}. \end{aligned}$$

Recall that $\{u_n\}_{n\in {\mathbb {N}}}$ is bounded in $W^{1,\mathcal {H}}_0(\Omega )$. Then, passing to the limit as $n\rightarrow \infty$ in the above estimate, we obtain

$$\begin{aligned}&(a_0+b_0\phi _{{\mathcal {H}}}^{\theta -1}(\nabla u_n)) \left[ \int _\Omega |\nabla u_n|^{p-2}\nabla u_n\cdot \nabla h\,\mathrm {d}x + \int _\Omega a(x)|\nabla u_n|^{q-2}\nabla u_n\cdot \nabla h\,\mathrm {d}x\right] \nonumber \\&\quad -\lambda \int _\Omega u_n^{-\gamma }h\,\mathrm {d}x - \int _\Omega u_n^{p^*-1}h\,\mathrm {d}x \ge o_n(1), \end{aligned}$$

(3.28)

for each nonnegative $h\in W^{1,\mathcal {H}}_0(\Omega )$, where we used the uniform boundedness from Lemma 3.8.

We aim to establish that (3.28) holds true for any arbitrary $h\in W^{1,\mathcal {H}}_0(\Omega )$. For this, we replace h in (3.28) by $(u_n+\varepsilon h)^+$ with $\varepsilon >0$ and $h\in W^{1,\mathcal {H}}_0(\Omega )$. Renaming as $h_{\varepsilon } = u_n+\varepsilon h$ and using (3.28), we get

$$\begin{aligned} o_n(1)&\le (a_0+b_0\phi _{{\mathcal {H}}}^{\theta -1}(\nabla u_n)) \\&\quad \left[ \int _\Omega |\nabla u_n|^{p-2}\nabla u_n\cdot \nabla h_\varepsilon ^+\,\mathrm {d}x + \int _\Omega a(x)|\nabla u_n|^{q-2}\nabla u_n\cdot \nabla h_\varepsilon ^+\,\mathrm {d}x\right] \\&\quad -\lambda \int _\Omega u_n^{-\gamma }h_\varepsilon ^+\,\mathrm {d}x - \int _\Omega u_n^{p^*-1}h_\varepsilon ^+ \,\mathrm {d}x \\&= (a_0+b_0\phi _{{\mathcal {H}}}^{\theta -1}(\nabla u_n))\\&\quad \left[ \int _\Omega |\nabla u_n|^{p-2}\nabla u_n\cdot \nabla h_\varepsilon ^-\,\mathrm {d}x + \int _\Omega a(x)|\nabla u_n|^{q-2}\nabla u_n\cdot \nabla h_\varepsilon ^-\,\mathrm {d}x\right] \\&\quad +(a_0+b_0\phi _{{\mathcal {H}}}^{\theta -1}(\nabla u_n)) \\&\quad \left[ \int _\Omega |\nabla u_n|^{p-2}\nabla u_n\cdot \nabla h_\varepsilon \,\mathrm {d}x + \int _\Omega a(x)|\nabla u_n|^{q-2}\nabla u_n\cdot \nabla h_\varepsilon \,\mathrm {d}x\right] \\&\quad -\lambda \int _\Omega u_n^{-\gamma }(h_\varepsilon + h_\varepsilon ^-)\,\mathrm {d}x - \int _\Omega u_n^{p^*-1}(h_\varepsilon +h_\varepsilon ^-)\,\mathrm {d}x\\&= \bigg [(a_0+b_0\phi _{{\mathcal {H}}}^{\theta -1}(\nabla u_n)) \left[ (\Vert u_n\Vert _p^p+ \Vert u_n\Vert _{q,a}^q)\right] -\lambda \int _\Omega |u_n|^{1-\gamma }\,\mathrm {d}x- \int _\Omega |u_n|^{p^*}\,\mathrm {d}x\bigg ]\\&\quad + \varepsilon \bigg \{(a_0+b_0\phi _{{\mathcal {H}}}^{\theta -1}(\nabla u_n)) \left[ \int _\Omega |\nabla u_n|^{p-2}\nabla u_n\cdot \nabla h\,\mathrm {d}x \right. \\&\quad \left. + \int _\Omega a(x)|\nabla u_n|^{q-2}\nabla u_n\cdot \nabla h\,\mathrm {d}x\right] \\&\quad -\lambda \int _\Omega u_n^{-\gamma }h\,\mathrm {d}x - \int _\Omega u_n^{p^*-1}h \,\mathrm {d}x \bigg \}-\lambda \int _\Omega u_n^{-\gamma } h_\varepsilon ^-\,\mathrm {d}x - \int _\Omega u_n^{p^*-1}h_\varepsilon ^-\,\mathrm {d}x\\&\quad +(a_0+b_0\phi _{{\mathcal {H}}}^{\theta -1}(\nabla u_n))\\&\quad \left[ \int _\Omega |\nabla u_n|^{p-2}\nabla u_n\cdot \nabla h_\varepsilon ^-\,\mathrm {d}x + \int _\Omega a(x)|\nabla u_n|^{q-2}\nabla u_n\cdot \nabla h_\varepsilon ^-\,\mathrm {d}x\right] . \end{aligned}$$

We define $\Omega _\varepsilon = \{x\in \Omega \,:\, u_n+\varepsilon h \le 0\}$. Using $u_n\in {\mathcal {N}}_\lambda$ and $\int _\Omega u_n^{-\gamma } h_\varepsilon ^-\,\mathrm {d}x\ge 0$ in the above estimate, we get

$$\begin{aligned} \begin{aligned} o_n(1)&\le \varepsilon \bigg \{(a_0+b_0\phi _{{\mathcal {H}}}^{\theta -1}(\nabla u_n)) \\ {}&\left[ \int _\Omega |\nabla u_n|^{p-2}\nabla u_n\cdot \nabla h\,\mathrm {d}x + \int _\Omega a(x)|\nabla u_n|^{q-2}\nabla u_n\cdot \nabla h\,\mathrm {d}x\right] \\&\qquad - \lambda \int _\Omega u_n^{-\gamma }h\,\mathrm {d}x - \int _\Omega u_n^{p^*-1}h \,\mathrm {d}x \bigg \}+ \int _{\Omega _\varepsilon } u_n^{p^*-1}h_\varepsilon \,\mathrm {d}x -(a_0+b_0\phi _{{\mathcal {H}}}^{\theta -1}(\nabla u_n))\\&\quad \left[ \int _{\Omega _\varepsilon } |\nabla u_n|^{p-2}\nabla u_n\cdot \nabla h_\varepsilon \,\mathrm {d}x + \int _{\Omega _\varepsilon } a(x)|\nabla u_n|^{q-2}\nabla u_n\cdot \nabla h_\varepsilon \,\mathrm {d}x\right] . \end{aligned} \end{aligned}$$

(3.29)

Note that

$$\begin{aligned} -\int _{\Omega _\varepsilon } |\nabla u_n|^{p-2}\nabla u_n\cdot \nabla h_\varepsilon \,\mathrm {d}x&= - \int _{\Omega _\varepsilon } |\nabla u_n|^{p-2}\nabla u_n\cdot \nabla (u_n+\varepsilon h)\,\mathrm {d}x\\&= -\int _{\Omega _\varepsilon }|\nabla u_n|^p\,\mathrm {d}x- \varepsilon \int _{\Omega _\varepsilon } |\nabla u_n|^{p-2}\nabla u_n\cdot \nabla h\,\mathrm {d}x\\&\le - \varepsilon \int _{\Omega _\varepsilon } |\nabla u_n|^{p-2}\nabla u_n\cdot \nabla h\,\mathrm {d}x \end{aligned}$$

and similarly,

$$\begin{aligned} - \int _{\Omega _\varepsilon } a(x)|\nabla u_n|^{q-2}\nabla u_n\cdot \nabla h_\varepsilon \,\mathrm {d}x \le - \varepsilon \int _{\Omega _\varepsilon } a(x)|\nabla u_n|^{q-2}\nabla u_n\cdot \nabla h\,\mathrm {d}x. \end{aligned}$$

Moreover, applying Hölder’s inequality and $u_n\le -\varepsilon h$ in $\Omega _\varepsilon$, we have

$$\begin{aligned} \left| \int _{\Omega _\varepsilon } u_n^{p^*-1}h_\varepsilon \,\mathrm {d}x\right|&\le \left| \int _{\Omega _\varepsilon } u_n^{p^*}\,\mathrm {d}x\right| + \epsilon \left| \int _{\Omega _\varepsilon } u_n^{p^*-1}|h|\,\mathrm {d}x\right| \\&\le \varepsilon ^{p^*}\int _{\Omega _\varepsilon } |h|^{p^*}\,\mathrm {d}x+ \varepsilon \left( \int _{\Omega _\varepsilon } u_n^{p^*}\,\mathrm {d}x\right) ^{\frac{p^*-1}{p^*}}\left( \int _{\Omega _\varepsilon } |h|^{p^*}\,\mathrm {d}x\right) ^{\frac{1}{p^*}}. \end{aligned}$$

Putting all these in (3.29), we infer that

$$\begin{aligned} \begin{aligned} o_n(1)&\le \varepsilon \bigg \{(a_0+b_0\phi _{{\mathcal {H}}}^{\theta -1}(\nabla u_n)) \left[ \int _\Omega |\nabla u_n|^{p-2}\nabla u_n\cdot \nabla h\,\mathrm {d}x + \int _\Omega a(x)|\nabla u_n|^{q-2}\nabla u_n\cdot \nabla h\,\mathrm {d}x\right] \\&\quad - \lambda \int _\Omega u_n^{-\gamma }h\,\mathrm {d}x - \int _\Omega u_n^{p^*-1}h \,\mathrm {d}x \bigg \}+ \varepsilon ^{p^*}\int _{\Omega _\varepsilon } |h|^{p^*}\,\mathrm {d}x\\&\quad + \varepsilon \left( \int _{\Omega _\varepsilon } u_n^{p^*}\,\mathrm {d}x\right) ^{\frac{p^*-1}{p^*}}\left( \int _{\Omega _\varepsilon } |h|^{p^*}\,\mathrm {d}x\right) ^{\frac{1}{p^*}}-\varepsilon (a_0+b_0\phi _{{\mathcal {H}}}^{\theta -1}(\nabla u_n))\\&\quad \left[ \int _{\Omega _\varepsilon } |\nabla u_n|^{p-2}\nabla u_n\cdot \nabla h\,\mathrm {d}x + \int _{\Omega _\varepsilon } a(x)|\nabla u_n|^{q-2}\nabla u_n\cdot \nabla h\,\mathrm {d}x\right] . \end{aligned} \end{aligned}$$

(3.30)

Since $|\Omega _\varepsilon |\rightarrow 0$ as $\varepsilon \rightarrow 0^+$ and by the boundedness of $\{u_n\}_{n\in {\mathbb {N}}}$ in $W^{1,\mathcal {H}}_0(\Omega )$, if we divide (3.30) by $\varepsilon >0$ and then pass to the limit as $\varepsilon \rightarrow 0^+$, we obtain

$$\begin{aligned} \begin{aligned}&(a_0+b_0\phi _{{\mathcal {H}}}^{\theta -1}(\nabla u_n)) \left[ \int _\Omega |\nabla u_n|^{p-2}\nabla u_n\cdot \nabla h\,\mathrm {d}x + \int _\Omega a(x)|\nabla u_n|^{q-2}\nabla u_n\cdot \nabla h\,\mathrm {d}x\right] \\&\quad -\lambda \int _\Omega u_n^{-\gamma }h\,\mathrm {d}x - \int _\Omega u_n^{p^*-1}h\,\mathrm {d}x \ge o_n(1), \end{aligned} \end{aligned}$$

(3.31)

as $n\rightarrow \infty$. By the arbitrariness of $h\in W^{1,\mathcal {H}}_0(\Omega )$, (3.31) actually implies (3.26) which completes the proof. $\square$

Now, we prove the compactness property of the energy functional $J_\lambda$ in a suitable range of $\lambda$. For this purpose, we set for any $\lambda >0$

$$\begin{aligned} c_\lambda := \alpha _2 - \alpha _1 \lambda ^{\frac{p^*}{p^*-1+\gamma }} \end{aligned}$$

where

$$\begin{aligned} \alpha _0:= \left( \frac{1}{q \theta } - \frac{1}{p^*}\right) , \qquad \alpha _1:= \frac{(p^*-1+\gamma ) |\Omega |}{p^*} \left( \frac{q\theta -1+\gamma }{q \theta (1-\gamma )}\right) ^{\frac{p^* }{p^*-1+\gamma }} \left( \frac{1-\gamma }{p^* \alpha _0}\right) ^\frac{1-\gamma }{p^*-1+\gamma } \end{aligned}$$

(3.32)

and

$$\begin{aligned} \alpha _2:= \alpha _0 \left( \frac{S b_0}{p^{\theta -1}}\right) ^\frac{p^*}{p^*-p} \left( S^{{p^*}} \left( \frac{b_0}{p^{\theta -1}}\right) ^{{p}}\right) ^{\frac{(\theta -1)p^*}{(p^*-p\theta )(p^*-p)}}. \end{aligned}$$

(3.33)

Also, for any $k \in {\mathbb {N}}$, let $T_k$ be the truncation defined by

$$\begin{aligned} T_k(t):= {\left\{ \begin{array}{ll} t &{} \text {if } |t| \le k, \\ k \displaystyle \frac{t}{|t|} &{} \text {if }|t| >k. \end{array}\right. } \end{aligned}$$

Proposition 3.10

Let hypotheses ($\hbox {h}_1$)-($\hbox {h}_2$) be satisfied, let $\lambda \in (0,\Lambda ^*)$ and let $\{u_n\}_{n\in {\mathbb {N}}}\subset \mathcal {N}_\lambda ^+$ be a sequence satisfying (3.8)–(3.9) and

$$\begin{aligned} J_\lambda (u_n) \rightarrow c < c_\lambda \quad \text {as } n \rightarrow \infty . \end{aligned}$$

(3.34)

Then $\{u_n\}_{n\in {\mathbb {N}}}$ possesses a strongly convergent subsequence in $W^{1,\mathcal {H}}_0(\Omega )$.

Proof

Fixing $k \in {\mathbb {N}}$ and taking $h= T_k(u_n-u_\lambda ) \in W^{1,\mathcal {H}}_0(\Omega )$ as a test function in (3.26), we get

$$\begin{aligned} \begin{aligned} o_n(1)&= (a_0+b_0\phi _{{\mathcal {H}}}^{\theta -1}(\nabla u_n)) \left[ \int _\Omega |\nabla u_n|^{p-2}\nabla u_n\nabla T_k(u_n-u_\lambda ) \,\mathrm {d}x\right. \\&\qquad \qquad \qquad \qquad \qquad \left. + \int _\Omega a(x)|\nabla u_n|^{q-2}\nabla u_n\nabla T_k(u_n-u_\lambda ) \,\mathrm {d}x\right] \\&\quad - \lambda \int _\Omega u_n^{-\gamma } \ T_k(u_n-u_\lambda ) \,\mathrm {d}x - \int _\Omega u_n^{p^*-1} T_k(u_n-u_\lambda ) \,\mathrm {d}x := I-J-K \quad \text {as } n \rightarrow \infty . \end{aligned} \end{aligned}$$

(3.35)

Using Young’s inequality, Propositions 2.1(iii)–(iv), 2.2(ii) and boundedness of the sequences $\{u_n\}_{n \in {\mathbb {N}}}$, $\{T_k(u_n-u_\lambda )\}_{n \in {\mathbb {N}}}$ in $W^{1,\mathcal {H}}_0(\Omega )$, we obtain

$$\begin{aligned} \begin{aligned} |J|&\le |I| + |K| + o_n(1) \\&\le (a_0+b_0\phi _{{\mathcal {H}}}^{\theta -1}(\nabla u_n)) \int _\Omega |\nabla u_n|^{p-1} |\nabla T_k(u_n-u_\lambda )| \,\mathrm {d}x \\ {}&\quad + \int _\Omega a(x)|\nabla u_n|^{q-1} |\nabla T_k(u_n-u_\lambda )| \,\mathrm {d}x \\&\quad + \int _\Omega |u_n|^{p^*-1} |T_k(u_n-u_\lambda )| \,\mathrm {d}x + o_n(1)\\&\le (a_0+b_0\phi _{{\mathcal {H}}}^{\theta -1}(\nabla u_n))\left( \rho _{\mathcal {H}}(\nabla u_n) + \rho _{\mathcal {H}}(\nabla T_k(u_n-u_\lambda ))\right) \\ {}&\quad + k \int _{\Omega } u_n^{p^*-1} \,\mathrm {d}x + o_n(1) \le C (1+k) \end{aligned} \end{aligned}$$

(3.36)

with a constant C independent of n and k, that is, the sequence $\{u_n^{-\gamma } \ T_k(u_n-u_\lambda )\}_{n \in {\mathbb {N}}}$ is uniformly integrable. Then, using (3.10) and Vitali’s convergence theorem, we get

$$\begin{aligned} \int _{\Omega } u_n^{-\gamma } \ T_k(u_n-u_\lambda ) \,\mathrm {d}x \rightarrow 0. \end{aligned}$$

By Hölder’s inequality, we observe that

$$\begin{aligned}{}[L^{\mathcal {H}}(\Omega )]^N \ni g \longmapsto \int _\Omega \left( |\nabla u_\lambda |^{p-2} + a(x)|\nabla u_\lambda |^{q-2} \right) \nabla u_\lambda \cdot g \,\mathrm {d}x \end{aligned}$$

is a bounded linear functional. From (3.10), we see that $\nabla T_k(u_n -u_\lambda ) \rightharpoonup 0$ in $[L^{\mathcal {H}}(\Omega )]^N$, so we can get

$$\begin{aligned} \lim _{n \rightarrow \infty } \int _\Omega \left( |\nabla u_\lambda |^{p-2} + a(x)|\nabla u_\lambda |^{q-2} \right) \nabla u_n \cdot \nabla T_k(u_n -u_\lambda ) \,\mathrm {d}x = 0. \end{aligned}$$

(3.37)

Let $\phi _{\mathcal {H}}(\nabla u_n) \rightarrow \beta := \frac{E_1}{p} + \frac{E_2}{q}$ as $n\rightarrow \infty$, where $E_1$ and $E_2$ are defined in (3.11). Thus, by using (3.36)–(3.37) in (3.35) and the fact that $a_0 \ge 0$, $b_0>0, \beta >0$, we get

$$\begin{aligned} (a_0 + b_0 \beta ^{\theta -1}) \limsup _{n \rightarrow \infty }&\bigg [ \int _{\Omega } \left( |\nabla u_n|^{p-2} \nabla u_n - |\nabla u_\lambda |^{p-2} \nabla u_\lambda \right) \cdot \nabla T_k(u_n-u_\lambda ) \,\mathrm {d}x \\&\quad + \int _{\Omega } a(x) \left( |\nabla u_n|^{q-2} \nabla u_n - |\nabla u_\lambda |^{q-2} \nabla u_\lambda \right) \cdot \nabla T_k(u_n-u_\lambda ) \,\mathrm {d}x \bigg ] \\&= \limsup _{n \rightarrow \infty } \int _\Omega u_n^{p^*-1} T_k(u_n-u_\lambda ) \,\mathrm {d}x \le C k. \end{aligned}$$

By Simon’s inequalities, see [24, formula (2.2)], we rewrite the above estimate as

$$\begin{aligned}&\limsup _{n \rightarrow \infty } \bigg [ \int _{\Omega } \left( |\nabla u_n|^{p-2} \nabla u_n - |\nabla u_\lambda |^{p-2} \nabla u_\lambda \right) \cdot \nabla T_k(u_n-u_\lambda ) \,\mathrm {d}x \bigg ] \nonumber \\&\le \frac{C k}{(a_0 + b_0 \beta ^{\theta -1})}. \end{aligned}$$

(3.38)

Set

$$\begin{aligned} s_n(x)= \left( |\nabla u_n|^{p-2} \nabla u_n - |\nabla u_\lambda |^{p-2} \nabla u_\lambda \right) \cdot \nabla (u_n-u_\lambda ). \end{aligned}$$

Note that $s_n(x) \ge 0$ a. e. in $\Omega$. We divide the set $\Omega$ by

$$\begin{aligned} E_n^k = \{x \in \Omega : |u_n(x) - u_\lambda (x)| \le k\} \ \ \text {and} \ \ F_n^k = \{x \in \Omega : |u_n(x) - u_\lambda (x)| > k\}, \end{aligned}$$

where $k, n \in {\mathbb {N}}$ are fixed. Let $\eta \in (0,1)$. Then, from the definition of $T_k$, Hölder’s inequality, (3.38) and the fact that $\lim _{n \rightarrow \infty } |F_n^k| =0$, we get

$$\begin{aligned} \limsup _{n \rightarrow \infty } \int _{\Omega } s_n^\eta \,\mathrm {d}x&\le \limsup _{n \rightarrow \infty } \left( \int _{E_n^k} s_n \,\mathrm {d}x\right) ^\eta |E_n^k|^{1-\eta } + \limsup _{n \rightarrow \infty } \left( \int _{F_n^k} s_n \,\mathrm {d}x\right) ^\eta |F_n^k|^{1-\eta } \,\mathrm {d}x.\\&\le \left( \frac{C k}{(a_0 + b_0 \beta ^{\theta -1})} \right) ^\eta |\Omega |^{1-\eta }. \end{aligned}$$

Letting $k \rightarrow 0^+$, we obtain that $s_n^\eta \rightarrow 0$ in $L^1(\Omega )$. Thus, we may assume that $s_n \rightarrow 0$ a. e. in $\Omega$ (up to a subsequence) which along with Simon’s inequalities [24, formula (2.2)] gives that

$$\begin{aligned} \nabla u_n \rightarrow \nabla u_\lambda \quad \text {a.\,e.\,in } \Omega . \end{aligned}$$

(3.39)

Let M be the nodal set of the weight function $a(\cdot )$ given by

$$\begin{aligned} M:= \{x \in \Omega : a(x)=0\}. \end{aligned}$$

Since, the sequences $\{|\nabla u_n|^{p-2} \nabla u_n\}_{n \in {\mathbb {N}}}$ and $\{|\nabla u_n|^{q-2} \nabla u_n\}_{n \in {\mathbb {N}}}$ are bounded in $L^{p'}(\Omega )$ and $L^{q'}(\Omega \setminus M, a(x) \,\mathrm {d}x)$, respectively, then by using (3.39) and [3, Proposition A.8], we conclude that

$$\begin{aligned} \int _{\Omega } |\nabla u_n|^{p-2} \nabla u_n \cdot \nabla u_\lambda = \Vert \nabla u_\lambda \Vert _p^p \end{aligned}$$

and

$$\begin{aligned} \int _{\Omega } a(x) |\nabla u_n|^{q-2} \nabla u_n \cdot \nabla u_\lambda = \int _{\Omega \setminus M} a(x) |\nabla u_n|^{q-2} \nabla u_n \cdot \nabla u_\lambda = \Vert \nabla u_\lambda \Vert _{q,a}^q. \end{aligned}$$

Furthermore, using (3.10), (3.39) and the Brezis-Lieb Lemma, we obtain

$$\begin{aligned} \begin{aligned}&\rho _{\mathcal {H}}(\nabla u_n) - \rho _{\mathcal {H}}(\nabla u_n- \nabla u_\lambda ) = \rho _{\mathcal {H}}(\nabla u_\lambda )+ o_n(1),\\&\Vert u_n\Vert _{p^*}^{p^*} - \Vert u_n-u_\lambda \Vert _{p^*}^{p^*} = \Vert u_\lambda \Vert _{p^*}^{p^*} + o_n(1) \end{aligned} \end{aligned}$$

(3.40)

as $n \rightarrow \infty$. Let $\Vert u_n -u_\lambda \Vert _{p^*} \rightarrow \ell$ for some $\ell \ge 0.$ Now, by taking $u_n-u_\lambda$ as a test function in (3.26), we get

$$\begin{aligned} o_n(1)&= (a_0+b_0\phi _{{\mathcal {H}}}^{\theta -1}(\nabla u_n)) \\&\left[ \int _\Omega |\nabla u_n|^{p-2}\nabla u_n \cdot \nabla (u_n-u_\lambda ) \,\mathrm {d}x + \int _\Omega a(x)|\nabla u_n|^{q-2}\nabla u_n \cdot \nabla (u_n-u_\lambda ) \,\mathrm {d}x\right] \\&\quad - \lambda \int _\Omega u_n^{-\gamma } \ (u_n-u_\lambda ) \,\mathrm {d}x - \int _\Omega u_n^{p^*-1} (u_n-u_\lambda ) \,\mathrm {d}x \\&= (a_0+b_0 \beta ^{\theta -1}) \left[ \rho _{\mathcal {H}}(\nabla u_n) - \rho _{\mathcal {H}}(\nabla u_\lambda ) + o_n(1) \right] - \Vert u_n\Vert _{p^*}^{p^*} + \Vert u_\lambda \Vert _{p^*}^{p^*} + o_n(1) \end{aligned}$$

as $n \rightarrow \infty .$ Hence, by (3.10) and (3.40) it follows that

$$\begin{aligned} (a_0+b_0 \beta ^{\theta -1}) \left[ \rho _{\mathcal {H}}(\nabla u_n) - \rho _{\mathcal {H}}(\nabla u_\lambda )\right] = \ell ^{p^*} + o_n(1) \quad \text {as } n \rightarrow \infty , \end{aligned}$$

(3.41)

which further gives

$$\begin{aligned} (a_0+b_0 \beta ^{\theta -1}) \lim _{n \rightarrow \infty } \left( \Vert \nabla u_n- \nabla u_\lambda \Vert _p^p + \Vert \nabla u_n- \nabla u_\lambda \Vert _{q,a}^q \right) \le \ell ^{p^*}. \end{aligned}$$

(3.42)

Now, we claim that $\ell =0$. Assume by contradiction that $\ell >0.$ By (3.1) and (3.42), we have

$$\begin{aligned} S a_0 \ell ^p \le S (a_0+b_0 \beta ^{\theta -1}) \ell ^p \le (a_0+b_0 \beta ^{\theta -1}) \lim _{n \rightarrow \infty } \Vert \nabla u_n - \nabla u_\lambda \Vert _p^p \le \ell ^{p^*}. \end{aligned}$$

(3.43)

Note that (3.42) implies that

$$\begin{aligned} (a_0+b_0 \beta ^{\theta -1}) (E_1+E_2 - \Vert \nabla u_\lambda \Vert ^p_p - \Vert \nabla u_\lambda \Vert ^q_{q,a}) \le \ell ^{p^*}. \end{aligned}$$

(3.44)

Using (3.43) in (3.44), we get

$$\begin{aligned} \begin{aligned} \left( \ell ^{p^*}\right) ^\frac{p^*-p}{p}&\ge (a_0+b_0 \beta ^{\theta -1})^\frac{p^*-p}{p} (E_1+E_2 - \Vert \nabla u_\lambda \Vert ^p_p - \Vert \nabla u_\lambda \Vert ^q_{q,a})^\frac{p^*-p}{p} \\&= (a_0+b_0 \beta ^{\theta -1})^\frac{p^*-p}{p} \lim _{n \rightarrow \infty } \left( \Vert \nabla u_n- \nabla u_\lambda \Vert _p^p + \Vert \nabla u_n- \nabla u_\lambda \Vert _{q,a}^q \right) ^\frac{p^*-p}{p} \\&\ge (a_0+b_0 \beta ^{\theta -1})^\frac{p^*-p}{p} \lim _{n \rightarrow \infty } \left( \Vert \nabla u_n- \nabla u_\lambda \Vert _p^p \right) ^\frac{p^*-p}{p}\\&\ge S^\frac{p^*-p}{p} (a_0+b_0 \beta ^{\theta -1})^\frac{p^*-p}{p} \ell ^{p^*-p}\\&\ge S^\frac{p^*}{p} (a_0+b_0 \beta ^{\theta -1})^\frac{p^*}{p}. \end{aligned} \end{aligned}$$

(3.45)

From (3.45) and (3.1), we obtain

$$\begin{aligned} E_1^{\frac{p^*-p}{p}}&\ge \left( E_1 - \Vert \nabla u_\lambda \Vert ^p_p\right) ^{\frac{p^*-p}{p}}\\ {}&= \left( \lim _{n \rightarrow \infty } \Vert \nabla u_n - \nabla u_\lambda \Vert ^p_p\right) ^{\frac{p^*-p}{p}} \ge S^{\frac{p^*-p}{p}} \ell ^{p^*-p}\\ {}&\ge S^{\frac{p^*}{p}} (a_0 + b_0 \beta ^{\theta -1}). \end{aligned}$$

This gives

$$\begin{aligned} E_1 \ge S^{\frac{p^*}{p^*-p}} (a_0 + b_0 \beta ^{\theta -1})^{\frac{p}{p^*-p}} \ge S^{\frac{p^*}{p^*-p}} \left( \frac{b_0}{p^{\theta -1}}\right) ^{\frac{p}{p^*-p}} E_1^{\frac{(\theta -1)p}{p^*-p}} \end{aligned}$$

and so we have

$$\begin{aligned} E_1 \ge \left[ S^{{p^*}} \left( \frac{b_0}{p^{\theta -1}}\right) ^{{p}}\right] ^{\frac{1}{p^*-p\theta }}. \end{aligned}$$

(3.46)

Combining (3.45) and (3.46), we obtain

$$\begin{aligned} \begin{aligned} \ell ^{p^*} \ge S^\frac{p^*}{p^*-p} (a_0+b_0 \beta ^{\theta -1})^\frac{p^*}{p^*-p}&\ge \left( \frac{S b_0}{p^{\theta -1}}\right) ^\frac{p^*}{p^*-p} E_1^{\frac{(\theta -1) p^*}{p^*-p}}\\&\ge \left( \frac{S b_0}{p^{\theta -1}}\right) ^\frac{p^*}{p^*-p} \left[ S^{{p^*}} \left( \frac{b_0}{p^{\theta -1}}\right) ^{{p}}\right] ^{\frac{(\theta -1)p^*}{(p^*-p\theta )(p^*-p)}}. \end{aligned} \end{aligned}$$

(3.47)

For any $n \in {\mathbb {N}}$, we have

$$\begin{aligned} J_\lambda (u_n) - \frac{1}{q \theta } \langle J_\lambda '(u_n), u_n \rangle&= a_0 \phi _{\mathcal {H}}(\nabla u_n) + \frac{b_0}{\theta } \phi ^{\theta }_{\mathcal {H}}(\nabla u_n) - \frac{1}{q \theta } m(\phi _\mathcal {H}(\nabla u_n)) \left\langle \mathcal {L}_{p,q}^{a}(u_n), u_n \right\rangle \\&\quad - \lambda \left( \frac{1}{1-\gamma } - \frac{1}{q \theta }\right) \int _{\Omega } u_n^{1-\gamma } \,\mathrm {d}x + \left( \frac{1}{q\theta } - \frac{1}{p^*}\right) \int _{\Omega } u_n^{p^*} \,\mathrm {d}x\\&\ge \left( \frac{1}{q\theta } - \frac{1}{p^*}\right) \Vert u_n\Vert _{p^*}^{p^*} - \lambda \left( \frac{1}{1-\gamma } - \frac{1}{q \theta }\right) \int _{\Omega } u_n^{1-\gamma } \,\mathrm {d}x. \end{aligned}$$

From this, as $n \rightarrow \infty$, by (3.47), (3.40), Hölder’s and Young’s inequality, we derive

$$\begin{aligned} c&= \lim _{n \rightarrow \infty } \left( J_\lambda (u_n) - \frac{1}{q\theta } \langle J_\lambda '(u_n), u_n \rangle \right) \\&\ge \alpha _0 \left( \ell ^{p^*} + \Vert u_\lambda \Vert _{p^* }^{p^*}\right) - \lambda \left( \frac{1}{1-\gamma } - \frac{1}{q \theta }\right) |\Omega |^{\frac{p^*-1+\gamma }{p^*}} \Vert u_\lambda \Vert _{p^*}^{1-\gamma }\\&\ge \alpha _0 \ell ^{p^*} - \alpha _1 \lambda ^{\frac{p^*}{p^*-1+\gamma }} \\&\ge \alpha _0 \left( \frac{S b_0}{p^{\theta -1}}\right) ^\frac{p^*}{p^*-p} \left[ S^{{p^*}} \left( \frac{b_0}{p^{\theta -1}}\right) ^{p\,}\right] ^{\frac{(\theta -1)p^*}{(p^*-p\theta )(p^*-p)}} - \alpha _1 \lambda ^{\frac{p^*}{p^*-1+\gamma }} = c_\lambda , \end{aligned}$$

where $\alpha _0$, $\alpha _1$ are defined in (3.32). The above estimates gives a contradiction to (3.34). Hence $\ell =0$ and using (3.41) and Proposition 2.1(v), we conclude the proof. $\square$

Remark 3.11

By taking $\lambda \in (0, \Lambda _*)$ with $\Lambda _*:= \left( \alpha _2 \alpha _1^{-1}\right) ^{\frac{p^* -1+\gamma }{p^*}}$ and $\alpha _1$, $\alpha _2$ are defined in (3.32) and (3.33) respectively, we have $c_\lambda >0.$

Proof

(Proof of Theorem 1.1) Fix $\lambda < \lambda ^*:= \min \{\Lambda ^*, \Lambda _*\}$. From Lemma 3.1(ii) and Ekeland’s variational principle there exists a minimizing sequence $\{u_n\}_{n \in {\mathbb {N}}} \in \mathcal {N}_\lambda ^+ \setminus \{0\}$ verifying (3.8), (3.9), (3.10) and (3.34) with $c= \Theta _\lambda ^+$. Hence, by combining Propositions 3.4 and 3.10 , we obtain $u_n \rightarrow u_\lambda$ strongly in $W^{1,\mathcal {H}}_0(\Omega )$ (up to a subsequence). This further implies that $u_\lambda \in \mathcal {N}_\lambda$ and by Lemma 3.7, we get $u_\lambda \in \mathcal {N}_\lambda ^+$ with $u_\lambda$ achieving $\Theta _\lambda ^+$ since $J_\lambda$ is continuous on $W^{1,\mathcal {H}}_0(\Omega ).$ Since $0 \not \in \mathcal {N}_\lambda ^+$ and $u_n \ge 0$ we have $u_\lambda \not \equiv 0$ and $u_\lambda \ge 0.$ Letting $n \rightarrow \infty$ in (3.26), we obtain that u satisfies $u_\lambda ^{-\gamma }\varphi \in L^1(\Omega )$ and

$$\begin{aligned} m(\phi _\mathcal {H}(\nabla u_\lambda )) \left\langle \mathcal {L}_{p,q}^{a}(u_\lambda ), \varphi \right\rangle = \lambda \int _{\Omega } u_\lambda ^{-\gamma }\varphi \,\mathrm {d}x +\int _\Omega u_\lambda ^{r-1} \varphi \,\mathrm {d}x \end{aligned}$$

for all $\varphi \in W^{1,\mathcal {H}}_0(\Omega )$. Finally, by using Proposition 3.4, Lemma 3.5 and by repeating the proof of [2, Proposition 4.3 and Proposition 4.4, Step 1], we obtain $u_\lambda > 0$ a. e. in $\Omega .$ $\square$