Kirchhoff problems with logarithmic double phase operator: Existence and multiplicity results

Abstract

In this paper, we focus on Kirchhoff type problems driven by a logarithmic double phase operator with variable exponents. Under very general assumptions on the nonlinearity and using variational tools, like the mountain pass theorem, we establish the existence of at least one nontrivial weak solution for the problem under consideration. Then, under different hypotheses on the reaction term, we are also able to derive a multiplicity result of solutions for our problem. We stress that in order to produce such a multiplicity result a key role is played by a variant of the symmetric mountain pass theorem.

Keywords

Logarithmic double phase operator Kirchhoff type problems variable exponents existence results multiplicity results

1. Introduction

Let $N ⩾ 2$ and $Ω \subseteq R^{N}$ be a bounded domain with Lipschitz boundary $\partial Ω$ . Given $r \in C (\bar{Ω})$ with $r (x) > 1$ for all $x \in \bar{Ω}$ we set

\begin{aligned} r^{-} := min_{x \in \bar{Ω}} r (x) and r^{+} := max_{x \in \bar{Ω}} r (x) . \end{aligned}

Also, let

C^{0, 1} (\bar{Ω})

be the space of all Lipschitz continuous functions

u : \bar{Ω} \to R

. In this paper, we explore the following Kirchhoff type problem:

\begin{aligned} - K [\int_{Ω} (\frac{1}{p (x)} | \nabla u |^{p (x)} + \frac{μ (x)}{q (x)} | \nabla u |^{q (x)} \log (e + | \nabla u |)) d x] div A (u) = f (x, u) \\ in Ω, \\ u = 0 on \partial Ω \end{aligned}

(1.1)

where e stands for Euler’s number and

div A

denotes the logarithmic double phase operator with variable exponents given for

u \in W^{1, H_{\log}} (Ω)

\begin{aligned} div A (u) : & = div (| \nabla u |^{p (x) - 2} \nabla u \\ + μ (x) [\log (e + | \nabla u |) + \frac{| \nabla u |}{q (x) (e + | \nabla u |)}] | \nabla u |^{q (x) - 2} \nabla u), \end{aligned}

(1.2)

being

W^{1, H_{\log}} (Ω)

the appropriate Musielak–Orlicz Sobolev space (see Section 2). Moreover, we suppose that the exponents, the weight function as well as the function

K

satisfy the following conditions:

(H1)

$p, q \in C^{0, 1} (\bar{Ω})$ are such that

\begin{aligned} 1 < p (x) < N and p (x) < q (x) < 2 q^{+} < (p^{*})^{-} ⩽ p^{*} (x) := \frac{N p (x)}{N - p (x)} \end{aligned}

for all

x \in \bar{Ω}

. In addition, the weight function is such that

\begin{aligned} 0 ⩽ μ (\cdot) \in L^{\infty} (Ω) ∖ {0} . \end{aligned}

(H2)

$K : [0, + \infty [\to [0, + \infty [$ is a continuous function verifying the following assumptions:

(i)

there exists $θ \in [1, \frac{(p^{*})^{-}}{2 q^{+}})$ such that

\begin{aligned} t K (t) ⩽ θ K (t) \end{aligned}

for all

t ⩾ 0

, being

K (t) = \int_{0}^{t} K (s) d s

;

(ii)

for any $τ > 0$ there exists $k = k (τ) > 0$ such that

\begin{aligned} K (t) ⩾ k for all t ⩾ τ . \end{aligned}

Example 1.1.

Let $k > 0$ and $θ \in [1, \frac{(p^{*})^{-}}{2 q^{+}})$ be fixed. Then, we remark that the function $K : [0, + \infty [\to [0, + \infty [$ defined by

\begin{aligned} K (t) := k + θ t^{θ - 1} for all t ⩾ 0 \end{aligned}

satisfies the hypothesis (H2).

Now, the main aim of this paper is twofold. First, we establish the existence for problem (1.1) of at least a nontrivial weak solution (see (3.1)) supposed that the nonlinearity f verifies some very general conditions (see hypothesis (H3) and Theorem 3.2). We stress that in order to do this we follow a variational approach based in particular on the use of the classic mountain pass theorem. Next, we present a multiplicity result of solutions for the problem under consideration (see Theorem 4.1). In order to provide such a result, we consider different but very general assumptions on the nonlinearity f (see hypotheses (H3)(ii) and (H4)). Also, we make again use of variational tools and more exactly of the Fountain theorem [25, Theorem 3.6] which is a variant of the symmetric mountain pass theorem.

We emphatize that distinctive feature of our work is the fact that it combines the logarithmic double phase operator given in (1.2) with a Kirchhoff term. To the best of our knowledge, there is no other work dealing with a Kirchhoff term along with such logarithmic double phase operator. Now, the operator introduced in (1.2) and the Musielak–Orlicz Sobolev space $W^{1, H_{\log}} (Ω)$ were studied by Arora–Crespo-Blanco–Winkert in [2]. There, the authors in addition proved multiplicity results for problems driven by the operator $d i v A$ but without the presence of a Kirchhoff coefficient and with a superlinear right-hand side.

As far as we know, a Kirchhoff type problem driven by the classic double phase operator with variable exponents was consider only recently by Ho–Winkert [15]. For Kirchhoff problems in double phase setting but with constant exponents we refer to the papers of Fiscella–Pinamonti [10] and Arora–Fiscella–Mukherjee–Winkert [3]. We point out that our work is closely related to the one of Fiscella–Pinamonti [10] where the authors analyze a similar problem to that under consideration here, producing existence and multiplicity results via analogous methods. But differently from us, they do not deal with a logarithmic operator. For some more studies on Kirchhoff differential problems, we refer to the papers of Cabanillas Lapa [7], Gupta–Dwivedi [13] and Ho–Kim–Zhang [14]. Next, we mention that nonlocal problems in the context of equations driven by the Laplacian or the p-Laplacian were studied by Alves–Figueiredo [1], Bueno–Ercole–Ferreira–Zumpano [6], Correa [8], Correa–Figueiredo [9], Gao–Kim–Zhen [12] and Sun et al. [22,23]. Further, existence results on degenerate and nondegenerate Kirchhoff problems can be find in the papers of Autuori–Pucci–Salvatori [4] and Xiang–Zhang–Rădulescu [26]. Also, we recall that $p (x)$ -Kirchhoff type problems with convection were studied by Vetro [24].

Finally, we underline that operators of type (1.2) are connected to integral functionals of the form

\begin{aligned} ν \to \int_{Ω} [| D ν |^{p (x)} + σ (x) | D ν |^{q (x)} \log (e + | D ν |)] d x . \end{aligned}

Such functionals in the case the exponents p, q are constant,

p = q

and

0 ⩽ σ (\cdot) \in C^{0, α} (Ω)

were investigated by Baroni–Colombo–Mingione [5] in order to establish the local Hölder continuity of the gradient of their local minimizers. We also observe that functionals with nearly linear growth of the form

\begin{aligned} ν \to \int_{Ω} | D ν | \log (1 + | D ν |) d x \end{aligned}

were considered by Fuchs–Mingione [11] and Marcellini–Papi [17]. Such type of functional appears in the context of theory of plasticity with logarithmic hardening, as one can see for example by Seregin–Frehse [21].

2. Mathematical background

This section is devoted to recall some basic elements from the theory of the variable exponent Lebesgue spaces and of the Musielak–Orlicz Sobolev spaces. The reader can find these topics in the books of Motreanu–Motreanu–Papageorgiou [18] and Musielak [19] and in the recent paper of Arora–Crespo-Blanco–Winkert [2].

Let $Ω \subseteq R^{N}$ ( $N ⩾ 2$ ) be a bounded domain with Lipschitz boundary and $M (Ω)$ be the set of all measurable functions $u : Ω \to R$ . Given $r \in C (\bar{Ω})$ with $r (x) > 1$ for all $x \in \bar{Ω}$ , we here denote with $L^{r (\cdot)} (Ω)$ the usual variable exponent Lebesgue space, that is,

\begin{aligned} L^{r (\cdot)} (Ω) = {u \in M (Ω) : \int_{Ω} | u |^{r (x)} d x < + \infty} \end{aligned}

equipped with the Luxemburg norm

\begin{aligned} ‖ u ‖_{r (\cdot)} = inf {α > 0 : \int_{Ω} {| \frac{u}{α} |}^{r (x)} d x ⩽ 1} . \end{aligned}

We stress that the above norm makes

L^{r (\cdot)} (Ω)

a separable, uniformly convex and hence reflexive Banach space whose dual space is given by

L^{r^{'} (\cdot)} (Ω)

, where

r^{'} (\cdot)

stands for the conjugate variable exponent to

r (\cdot)

, that is, we have

\begin{aligned} \frac{1}{r (x)} + \frac{1}{r^{'} (x)} = 1 for all x \in \bar{Ω} . \end{aligned}

Next, we recall that the Hölder-type inequality

\begin{aligned} \int_{Ω} | u v | d x ⩽ [\frac{1}{r^{-}} + \frac{1}{(r^{'})^{-}}] ‖ u ‖_{r (\cdot)} ‖ v ‖_{r^{'} (\cdot)} \end{aligned}

holds for all

u \in L^{r (\cdot)} (Ω)

and all

v \in L^{r^{'} (\cdot)} (Ω)

. Also, given

r_{1}, r_{2} \in C (\bar{Ω})

, with

1 < r_{1} (x) ⩽ r_{2} (x)

for all

x \in \bar{Ω}

, we have the continuous embedding

\begin{aligned} L^{r_{2} (\cdot)} (Ω) ↪ L^{r_{1} (\cdot)} (Ω) . \end{aligned}

Now, we assume that hypothesis (H1) is satisfied. Thus, we focus on the nonlinear function

H_{\log} : Ω \times [0, + \infty) \to [0, + \infty)

defined by

\begin{aligned} H_{\log} (x, t) = t^{p (x)} + μ (x) t^{q (x)} \log (e + t) \end{aligned}

for all

x \in Ω

and for all

t ⩾ 0

. We note that

H_{\log} (\cdot, t)

is measurable for all

t ⩾ 0

H_{\log} (x, t) > 0

for all

t > 0

and

H_{\log} (x, 0) = 0

. In addition, we can easily see that the function

H_{\log}

satisfies the

Δ_{2}

-condition. On the base of this, we can consider the corresponding Musielak–Orlicz space

L^{H_{\log}} (Ω)

which is given by

\begin{aligned} L^{H_{\log}} (Ω) = {u \in M (Ω) : ϱ_{H_{\log}} (u) < + \infty} \end{aligned}

endowed with the Luxemburg norm

\begin{aligned} ‖ u ‖_{H_{\log}} := inf {β > 0 : ϱ_{H_{\log}} (\frac{u}{β}) ⩽ 1}, \end{aligned}

where

ϱ_{H_{\log}} (\cdot)

denotes the associated modular defined by

\begin{aligned} ϱ_{H_{\log}} (u) := \int_{Ω} H_{\log} (x, | u |) d x = \int_{Ω} (| u |^{p (x)} + μ (x) | u |^{q (x)} \log (e + | u |)) d x . \end{aligned}

We point out that

L^{H_{\log}} (Ω)

is a separable, reflexive Banach space whose norm

‖ \cdot ‖_{H_{\log}}

is closely related to the modular

ϱ_{H_{\log}}

, see [2, Proposition 3.4]. Next, we recall some relations between

‖ \cdot ‖_{H_{\log}}

and

ϱ_{H_{\log}}

Proposition 2.1.
Let hypothesis (H1) be satisfied. Then the following hold: (i)
$‖ u ‖_{H_{\log}} < 1$ (resp. $> 1$ , $= 1$ ) if and only if $ϱ_{H_{\log}} (u) < 1$ (resp. $> 1$ , $= 1$ );
(ii)
$min {‖ u ‖_{H_{\log}}^{p^{-}}, ‖ u ‖_{H_{\log}}^{q^{+} + η}} ⩽ ϱ_{H_{\log}} (u) ⩽ max {‖ u ‖_{H_{\log}}^{p^{-}}, ‖ u ‖_{H_{\log}}^{q^{+} + η}}$ , where $η = \frac{e}{e + s_{0}}$ with $s_{0}$ being the only positive solution of $s_{0} = e \log (e + s_{0})$ ;
(iii)
$‖ u ‖_{H_{\log}} \to 0$ if and only if $ϱ_{H_{\log}} (u) \to 0$ ;
(iv)
$‖ u ‖_{H_{\log}} \to + \infty$ if and only if $ϱ_{H_{\log}} (u) \to + \infty$ .

Using $L^{H_{\log}} (Ω)$ , we define the Musielak–Orlicz Sobolev space $W^{1, H_{\log}} (Ω)$ by
$\begin{aligned} W^{1, H_{\log}} (Ω) = {u \in L^{H_{\log}} (Ω) : | \nabla u | \in L^{H_{\log}} (Ω)} \end{aligned}$
furnished with the norm
$\begin{aligned} ‖ u ‖_{1, H_{\log}} := ‖ u ‖_{H_{\log}} + ‖ \nabla u ‖_{H_{\log}}, \end{aligned}$
where as usual $‖ \nabla u ‖_{H_{\log}} := ‖ | \nabla u | ‖_{H_{\log}}$ . Then, we write $W_{0}^{1, H_{\log}} (Ω)$ by the completion of $C_{0}^{\infty} (Ω)$ in $W^{1, H_{\log}} (Ω)$ . We emphatize that $W^{1, H_{\log}} (Ω) ~and~ W_{0}^{1, H_{\log}} (Ω)$ are separable, reflexive Banach spaces satisfying the following embeddings, see [2, Propositions 3.6, 3.7 and 3.9].
Proposition 2.2.
Let hypothesis (H1) be satisfied. Then the following hold: (i)
$W^{1, H_{\log}} (Ω) ↪ L^{r (\cdot)} (Ω)$ and $W_{0}^{1, H_{\log}} (Ω) ↪ L^{r (\cdot)} (Ω)$ are compact for $r \in C (\bar{Ω})$ with $1 ⩽ r (x) < p^{} (x)$ for all* $x \in \bar{Ω}$ ;
(ii)
$W^{1, H_{\log}} (Ω) ↪ L^{H_{\log}} (Ω)$ is compact and there exists a constant $a > 0$ such that
$\begin{aligned} ‖ u ‖_{H_{\log}} ⩽ a ‖ \nabla u ‖_{H_{\log}} for all u \in W_{0}^{1, H_{\log}} (Ω) . \end{aligned}$

So, in accordance with Proposition 2.2 (ii) we consider in $W_{0}^{1, H_{\log}} (Ω)$ the equivalent norm given by
$\begin{aligned} ‖ u ‖ := ‖ \nabla u ‖_{H_{\log}} for all u \in W_{0}^{1, H_{\log}} (Ω) . \end{aligned}$
In the sequel, we denote by $A : W_{0}^{1, H_{\log}} (Ω) \to [W_{0}^{1, H_{\log}} (Ω)]^{}$ the nonlinear operator defined by
$\begin{aligned} ⟨ A (u), v ⟩ : & = \int_{Ω} | \nabla u |^{p (x) - 2} \nabla u \cdot \nabla v d x \\ + μ (x) [\log (e + | \nabla u |) + \frac{| \nabla u |}{q (x) (e + | \nabla u |)}] | \nabla u |^{q (x) - 2} \nabla u \cdot \nabla v d x \end{aligned}$
(2.1)
for all $u, v \in W_{0}^{1, H_{\log}} (Ω)$ , being $⟨ \cdot, \cdot ⟩$ the dual pairing between $W_{0}^{1, H_{\log}} (Ω)$ and its dual space. Such operator is characterized by remarkable properties, see [2, Theorem 4.4]. In particular, we have the following result.
Proposition 2.3.
Let hypothesis* (H1) be satisfied. Then, the operator $A$ is bounded (that is, it maps bounded sets into bounded sets), continuous, strictly monotone, coercive, that is,
$\begin{aligned} lim_{‖ u ‖ \to + \infty} \frac{⟨ A (u), u ⟩}{‖ u ‖} = + \infty, \end{aligned}$
and also of $(S_{+})$ -type, that is,
$\begin{aligned} u_{n} ⇀ u i n W_{0}^{1, H_{\log}} (Ω) a n d \underset{n \to + \infty}{\lim \sup} ⟨ A (u_{n}), u_{n} - u ⟩ ⩽ 0 \end{aligned}$
imply $u_{n} \to u ~in~ W_{0}^{1, H_{\log}} (Ω)$ .

Now, we recall that a functional $Ψ : W_{0}^{1, H_{\log}} (Ω) \to R$ satisfies the Palais–Smale condition if any sequence ${u_{n}}_{n \in N} \subset W_{0}^{1, H_{\log}} (Ω)$ such that
$\begin{aligned} {Ψ (u_{n})}_{n \in N} \subset R is bounded and \\ Ψ^{'} (u_{n}) \to 0 i n {[W_{0}^{1, H_{\log}} (Ω)]}^{} ~as~ n \to + \infty \end{aligned}$
admits a convergent subsequence in* $W_{0}^{1, H_{\log}} (Ω)$ .

Lastly, we fix some notation which we need later. For any $s \in R$ we put $s_{\pm} := max {\pm s, 0}$ which means $s = s_{+} - s_{-}$ and $| s | = s_{+} + s_{-}$ . Moreover, for any function $u : Ω \to R$ we set $u_{\pm} (\cdot) := [u (\cdot)]_{\pm}$ . Also, with the purpose to lighten the notation, from now on we will use C by denoting a generic constant, which may change from line to line, but does not depend on the crucial quantities.
3. Existence of nontrivial weak solutions

In this section, we present our first existence result. More exactly, we show that problem (1.1) admits at least a nontrivial weak solution supposed that the nonlinearity f verifies some very general conditions. In particular, we here make the following assumptions on f.

(H3)
Let η and θ be as given in Proposition 2.1 and hypothesis (H2)(i), respectively. Then, $f : Ω \times R \to R$ is a Carathéodory function satisfying the following conditions:
(i)
there exists $l \in ((q^{+} + η) θ, (p^{})^{-})$ such that for all $ε > 0$ it is possible to find $δ_{ε} > 0$ so that it results
$\begin{aligned} | f (x, t) | ⩽ (q^{+} + η) θ ε | t |^{(q^{+} + η) θ - 1} + l δ_{ε} | t |^{l - 1} \end{aligned}$
for a.a. $x \in Ω$ and for all $t \in R$ ;
(ii)
there exist $m \in (2 q^{+} θ, (p^{})^{-})$ and $t_{0} ⩾ 0$ such that
$\begin{aligned} c ⩽ m F (x, t) ⩽ t f (x, t) \end{aligned}$
for some $c > 0$ , for a.a. $x \in Ω$ and for any $| t | ⩾ t_{0}$ , with $F (x, t) = \int_{0}^{t} f (x, s) d s$ .
Example 3.1.
Let $θ \in [1, \frac{(p^{})^{-}}{2 q^{+}})$ and* $l \in (2 q^{+} θ, (p^{})^{-})$ be fixed. Then, the odd function* $f : Ω \times R \to R$ defined by
$\begin{aligned} f (x, t) := l t^{l - 1} f o r a . a . x \in Ω and for all t ⩾ 0 \end{aligned}$
satisfies all the assumptions in (H3). We in fact stress that condition (H3)(i) holds with
$\begin{aligned} δ_{ε} = 1 for all ε > 0. \end{aligned}$
While, condition (H3)(ii) is verified with
$\begin{aligned} t_{0} = 1 a n d m \in (2 q^{+} θ, l) . \end{aligned}$

We recall that $u \in W_{0}^{1, H_{\log}} (Ω)$ is a weak solution for problem (1.1) if
$\begin{aligned} K [\int_{Ω} (\frac{1}{p (x)} | \nabla u |^{p (x)} + \frac{μ (x)}{q (x)} | \nabla u |^{q (x)} \log (e + | \nabla u |)) d x] \\ \times \int_{Ω} A (u) \cdot \nabla v d x = \int_{Ω} f (x, u) v d x \end{aligned}$
(3.1)
is verified for all $v \in W_{0}^{1, H_{\log}} (Ω)$ . Now, our strategy in order to establish the existence of a nontrivial weak solution for problem (1.1) is to use the classic mountain pass theorem. To this purpose, we consider the functional $Φ : W_{0}^{1, H_{\log}} (Ω) \to R$ defined by
$\begin{aligned} Φ (u) := K [\int_{Ω} (\frac{1}{p (x)} | \nabla u |^{p (x)} + \frac{μ (x)}{q (x)} | \nabla u |^{q (x)} \log (e + | \nabla u |)) d x] - \int_{Ω} F (x, u) d x \end{aligned}$
(3.2)
for all $u \in W_{0}^{1, H_{\log}} (Ω)$ . We emphatize that Φ is a $C^{1}$ -functional with derivative given by
$\begin{aligned} ⟨ Φ^{'} (u), v ⟩ & = K [\int_{Ω} (\frac{1}{p (x)} | \nabla u |^{p (x)} + \frac{μ (x)}{q (x)} | \nabla u |^{q (x)} \log (e + | \nabla u |)) d x] \\ \times \int_{Ω} A (u) \cdot \nabla v d x - \int_{Ω} f (x, u) v d x \end{aligned}$
(3.3)
for all $u, v \in W_{0}^{1, H_{\log}} (Ω)$ . Consequently, we have that the critical points of Φ are the weak solutions of problem (1.1).

Finally, we are in the position to give our first existence result. Theorem 3.2.
Let hypotheses (H1), (H2) and (H3) be satisfied. Then, problem (1.1) admits at least one nontrivial weak solution in $W_{0}^{1, H_{\log}} (Ω)$ .
Proof.
Our idea is to show that the functional Φ satisfies the geometric features of the classic mountain pass theorem. Done this in fact we are able to apply such theorem in order to obtain the existence of a nontrivial critical value of Φ and consequently of a nontrivial weak solution for problem (1.1) in $W_{0}^{1, H_{\log}} (Ω)$ (see (3.1) and (3.3)). We are going to prove the claim in four steps.

Step 1. As first step, we show that there exist $ρ \in (0, 1]$ and $ζ := ζ (ρ) > 0$ such that
$\begin{aligned} Φ (u) ⩾ ζ for all u \in W_{0}^{1, H_{\log}} (Ω) ~with~ ‖ u ‖ = ρ . \end{aligned}$
We observe that, according to hypothesis (H3)(i), for any $ε > 0$ it is possible to find a $δ_{ε} > 0$ such that
$\begin{aligned} | F (x, t) | ⩽ ε | t |^{(q^{+} + η) θ} + δ_{ε} | t |^{l} \end{aligned}$
for a.a. $x \in Ω$ and for all $t \in R$ . Also, using the assumptions on the function $K$ (see hypotheses (H2)), we derive that
$\begin{aligned} K (t) ⩾ K (1) t^{θ} for all t \in [0, 1] . \end{aligned}$
Finally, from Proposition 2.1(ii) we know that for all $u \in W_{0}^{1, H_{\log}} (Ω)$ with $‖ u ‖ ⩽ 1$ it results
$\begin{aligned} ‖ u ‖^{q^{+} + η} ⩽ ϱ_{H_{\log}} (\nabla u) ⩽ ‖ u ‖^{p^{-}} ⩽ 1 \end{aligned}$
which implies
$\begin{aligned} \int_{Ω} (\frac{1}{p (x)} | \nabla u |^{p (x)} + \frac{μ (x)}{q (x)} | \nabla u |^{q (x)} \log (e + | \nabla u |)) d x < 1 \end{aligned}$
being
$\begin{aligned} \int_{Ω} (\frac{1}{p (x)} | \nabla u |^{p (x)} + \frac{μ (x)}{q (x)} | \nabla u |^{q (x)} \log (e + | \nabla u |)) d x ⩽ \frac{1}{p^{-}} ϱ_{H_{\log}} (\nabla u) . \end{aligned}$
Then, taking into account of all this, we have that
$\begin{aligned} Φ (u) & ⩾ K (1) {[\int_{Ω} (\frac{1}{p (x)} | \nabla u |^{p (x)} + \frac{μ (x)}{q (x)} | \nabla u |^{q (x)} \log (e + | \nabla u |)) d x]}^{θ} \\ - \int_{Ω} ε | u |^{(q^{+} + η) θ} d x - \int_{Ω} δ_{ε} | u |^{l} d x \\ ⩾ \frac{K (1)}{(q^{+})^{θ}} ‖ u ‖^{(q^{+} + η) θ} - ε ‖ u ‖_{(q^{+} + η) θ}^{(q^{+} + η) θ} - δ_{ε} ‖ u ‖_{l}^{l} . \end{aligned}$
We recall that $(q^{+} + η) θ < 2 q^{+} θ < (p^{})^{-}$ and $l < (p^{})^{-}$ due to hypotheses (H2)(i) and (H3)(i), respectively. Therefore, from Proposition 2.2(i) we have that both the embeddings $W_{0}^{1, H_{\log}} (Ω) ↪ L^{(q^{+} + η) θ} (Ω)$ and $W_{0}^{1, H_{\log}} (Ω) ↪ L^{l} (Ω)$ are compact. This permits us to deduce that
$\begin{aligned} Φ (u) & ⩾ \frac{K (1)}{(q^{+})^{θ}} ‖ u ‖^{(q^{+} + η) θ} - ε C ‖ u ‖^{(q^{+} + η) θ} - δ_{ε} C ‖ u ‖^{l} \\ = (\frac{K (1)}{(q^{+})^{θ}} - ε C) ‖ u ‖^{(q^{+} + η) θ} - δ_{ε} C ‖ u ‖^{l} \\ = [(\frac{K (1)}{(q^{+})^{θ}} - ε C) - δ_{ε} C ‖ u ‖^{l - (q^{+} + η) θ}] ‖ u ‖^{(q^{+} + η) θ} \end{aligned}$
for some $C > 0$ , where $l - (q^{+} + η) θ > 0$ due to hypothesis (H3)(i). Now, if we take $ε > 0$ small enough so that it results
$\begin{aligned} \frac{K (1)}{(q^{+})^{θ}} - ε C > 0, \end{aligned}$
for any $u \in W_{0}^{1, H_{\log}} (Ω)$ with
$\begin{aligned} ‖ u ‖ = ρ \in (0, min {1, {[\frac{1}{δ_{ε} C} (\frac{K (1)}{(q^{+})^{θ}} - ε C)]}^{\frac{1}{l - (q^{+} + η) θ}}}) \end{aligned}$
we have
$\begin{aligned} Φ (u) ⩾ [(\frac{K (1)}{(q^{+})^{θ}} - ε C) - δ_{ε} C ρ^{l - (q^{+} + η) θ}] ρ^{(q^{+} + η) θ} := ζ > 0, \end{aligned}$
that is, the assertion holds.

Step 2. Our goal is now in proving that there exists $w \in W_{0}^{1, H_{\log}} (Ω)$ such that
$\begin{aligned} Φ (w) < 0 and ‖ w ‖ > 1. \end{aligned}$
We stress that, according to hypotheses (H3), we can find $b_{1} > 0$ and $b_{2} ⩾ 0$ so that
$\begin{aligned} F (x, t) ⩾ b_{1} | t |^{m} - b_{2} \end{aligned}$
(3.4)
holds for a.a. $x \in Ω$ and for all $t \in R$ . Also, integrating (H2)(i) we see that for all $ε > 0$ it results
$\begin{aligned} K (t) ⩽ \frac{K (ε)}{ε^{θ}} t^{θ} whenever t ⩾ ε . \end{aligned}$
(3.5)
Next, thanks to Proposition 2.1(ii) we know that for all $u \in W_{0}^{1, H_{\log}} (Ω)$ with $‖ u ‖ ⩾ (q^{+})^{\frac{1}{p^{-}}} > 1$ it holds
$\begin{aligned} \int_{Ω} (\frac{1}{p (x)} | \nabla u |^{p (x)} + \frac{μ (x)}{q (x)} | \nabla u |^{q (x)} \log (e + | \nabla u |)) d x & ⩾ \frac{1}{q^{+}} ϱ_{H_{\log}} (\nabla u) \\ ⩾ \frac{1}{q^{+}} ‖ u ‖^{p^{-}} ⩾ 1. \end{aligned}$
(3.6)
Keeping all this in mind, for $v \in W_{0}^{1, H_{\log}} (Ω)$ with $‖ v ‖ = 1$ and $t ⩾ (q^{+})^{\frac{1}{p^{-}}}$ we get
$\begin{aligned} Φ (t v) & ⩽ K (1) {[\int_{Ω} (\frac{1}{p (x)} | \nabla (t v) |^{p (x)} + \frac{μ (x)}{q (x)} | \nabla (t v) |^{q (x)} \log (e + | \nabla (t v) |)) d x]}^{θ} \\ - b_{1} \int_{Ω} | t v |^{m} d x + b_{2} \int_{Ω} d x \\ ⩽ \frac{K (1)}{p^{-}} t^{(q^{+} + 1) θ} {[\int_{Ω} (| \nabla v |^{p (x)} + μ (x) | \nabla v |^{q (x)} \log (e + | \nabla v |)) d x]}^{θ} \\ - b_{1} t^{m} ‖ v ‖_{m}^{m} + b_{2} | Ω | \\ = \frac{K (1)}{p^{-}} t^{(q^{+} + 1) θ} {[ϱ_{H_{\log}} (\nabla v)]}^{θ} - b_{1} t^{m} ‖ v ‖_{m}^{m} + b_{2} | Ω | . \end{aligned}$
Now, we remark that $‖ v ‖ := ‖ \nabla v ‖_{H_{\log}} = 1$ implies $ϱ_{H_{\log}} (\nabla v) = 1$ in accordance with Proposition 2.1(i). Also, as $m < (p^{})^{-}$ due to hypothesis (H3)(ii), from Proposition 2.2 (i) we deduce that the embedding $W_{0}^{1, H_{\log}} (Ω) ↪ L^{m} (Ω)$ is compact. Consequently, we can write
$\begin{aligned} Φ (t v) ⩽ \frac{K (1)}{p^{-}} t^{(q^{+} + 1) θ} - b_{1} C t^{m} + b_{2} | Ω | \end{aligned}$
for some $C > 0$ and hence, as $m > (q^{+} + 1) θ$ due to hypothesis (H3)(ii), we derive that
$\begin{aligned} Φ (t v) \to - \infty as t \to + \infty . \end{aligned}$
Then, for $\bar{t} > 0$ large enough we have that $w = \bar{t} v \in W_{0}^{1, H_{\log}} (Ω)$ is such that $‖ w ‖ > 1$ and in addition $Φ (w) < 0$ , that is, the claim is proved.

Step 3. We now claim that the functional Φ satisfies the Palais–Smale condition. Thus, we consider a sequence ${u_{n}}_{n \in N} \subset W_{0}^{1, H_{\log}} (Ω)$ such that
$\begin{aligned} {Φ (u_{n})}_{n \in N} \subset R is bounded and Φ^{'} (u_{n}) \to 0 in {[W_{0}^{1, H_{\log}} (Ω)]}^{} as n \to + \infty . \end{aligned}$
(3.7)
We affirm that the sequence ${u_{n}}_{n \in N}$ is bounded. In fact, if we suppose that it is unbounded we can find $n_{0} \in N$ and a subsequence of ${u_{n}}_{n \in N}$ , here namely again ${u_{n}}_{n \in N}$ , so that it results
$\begin{aligned} | | u_{n} | | \to + \infty as n \to + \infty and | | u_{n} | | ⩾ {(q^{+})}^{\frac{1}{p^{-}}} for all n ⩾ n_{0} . \end{aligned}$
(3.8)
Then, with a view to hypothesis (H2)(i), we see that
$\begin{aligned} Φ (u_{n}) - \frac{1}{m} ⟨ Φ^{'} (u_{n}), u_{n} ⟩ \\ = K [\int_{Ω} (\frac{1}{p (x)} | \nabla u_{n} |^{p (x)} + \frac{μ (x)}{q (x)} | \nabla u_{n} |^{q (x)} \log (e + | \nabla u_{n} |)) d x] - \int_{Ω} F (x, u_{n}) d x \\ - \frac{1}{m} K [\int_{Ω} (\frac{1}{p (x)} | \nabla u_{n} |^{p (x)} + \frac{μ (x)}{q (x)} | \nabla u_{n} |^{q (x)} \log (e + | \nabla u_{n} |)) d x] \\ \times \int_{Ω} (| \nabla u_{n} |^{p (x)} + μ (x) [\log (e + | \nabla u_{n} |) + \frac{| \nabla u_{n} |}{q (x) (e + | \nabla u_{n} |)}] | \nabla u_{n} |^{q (x)}) d x \\ + \frac{1}{m} \int_{Ω} f (x, u_{n}) u_{n} d x \\ ⩾ \frac{1}{q^{+} θ} K [\int_{Ω} (\frac{1}{p (x)} | \nabla u_{n} |^{p (x)} + \frac{μ (x)}{q (x)} | \nabla u_{n} |^{q (x)} \log (e + | \nabla u_{n} |)) d x] \\ \times \int_{Ω} (| \nabla u_{n} |^{p (x)} + μ (x) | \nabla u_{n} |^{q (x)} \log (e + | \nabla u_{n} |)) d x \\ - \frac{1}{m} K [\int_{Ω} (\frac{1}{p (x)} | \nabla u_{n} |^{p (x)} + \frac{μ (x)}{q (x)} | \nabla u_{n} |^{q (x)} \log (e + | \nabla u_{n} |)) d x] \\ \times \int_{Ω} (| \nabla u_{n} |^{p (x)} + μ (x) [\log (e + | \nabla u_{n} |) + \frac{| \nabla u_{n} |}{q (x) (e + | \nabla u_{n} |)}] | \nabla u_{n} |^{q (x)}) d x \\ - \int_{Ω} (F (x, u_{n}) - \frac{1}{m} f (x, u_{n}) u_{n}) d x \\ ⩾ (\frac{1}{q^{+} θ} - \frac{1}{m}) K [\int_{Ω} (\frac{1}{p (x)} | \nabla u_{n} |^{p (x)} + \frac{μ (x)}{q (x)} | \nabla u_{n} |^{q (x)} \log (e + | \nabla u_{n} |)) d x] \\ \times ϱ_{H_{\log}} (\nabla u_{n}) - \int_{Ω} (F (x, u_{n}) - \frac{1}{m} f (x, u_{n}) u_{n}) d x \\ - \frac{1}{m} K [\int_{Ω} (\frac{1}{p (x)} | \nabla u_{n} |^{p (x)} + \frac{μ (x)}{q (x)} | \nabla u_{n} |^{q (x)} \log (e + | \nabla u_{n} |)) d x] \\ \times \int_{Ω} μ (x) \frac{| \nabla u_{n} |}{q (x) (e + | \nabla u_{n} |)} | \nabla u_{n} |^{q (x)} d x \\ ⩾ (\frac{1}{q^{+} θ} - \frac{2}{m}) K [\int_{Ω} (\frac{1}{p (x)} | \nabla u_{n} |^{p (x)} + \frac{μ (x)}{q (x)} | \nabla u_{n} |^{q (x)} \log (e + | \nabla u_{n} |)) d x] \\ \times ϱ_{H_{\log}} (\nabla u_{n}) - \int_{Ω_{n}} {[F (x, u_{n}) - \frac{1}{m} f (x, u_{n}) u_{n}]}_{+} d x \end{aligned}$
where $Ω_{n} := {x \in Ω : | u_{n} (x) | ⩽ t_{0}}$ , being $t_{0}$ as given in hypothesis (H3)(ii), and
$\begin{aligned} \frac{1}{q^{+} θ} - \frac{2}{m} > 0 due to hypothesis (H 3) (ii) . \end{aligned}$
Now, we put
$\begin{aligned} B := | Ω | sup_{x \in Ω, | t | ⩽ t_{0}} {[F (x, t) - \frac{1}{m} f (x, t) t]}_{+} \end{aligned}$
where $| Ω |$ denotes the Lebesgue measure of Ω in $R^{N}$ . We stress that from hypothesis (H3)(i) we can easily deduce that $B < + \infty$ . Moreover, thanks to hypothesis (H2)(ii) we know that there exists $k > 0$ such that
$\begin{aligned} K (t) ⩾ k whenever t ⩾ 1. \end{aligned}$
(3.9)
Thus, as
$\begin{aligned} \int_{Ω} (\frac{1}{p (x)} | \nabla u_{n} |^{p (x)} + \frac{μ (x)}{q (x)} | \nabla u_{n} |^{q (x)} \log (e + | \nabla u_{n} |)) d x ⩾ 1 \end{aligned}$
by (3.6) and (3.8), using (3.9) and Proposition 2.1(ii), we derive that
$\begin{aligned} Φ (u_{n}) - \frac{1}{m} ⟨ Φ^{'} (u_{n}), u_{n} ⟩ & ⩾ (\frac{1}{q^{+} θ} - \frac{2}{m}) k ‖ u_{n} ‖^{p^{-}} - B . \end{aligned}$
Based on this, in accordance with (3.7), we can affirm that there exist $b_{3}, b_{4} > 0$ such that
$\begin{aligned} b_{3} + b_{4} ‖ u_{n} ‖ ⩾ (\frac{1}{q^{+} θ} - \frac{2}{m}) k ‖ u_{n} ‖^{p^{-}} - B for all n \in N . \end{aligned}$
Clearly, this is in contradiction with (3.8) being $p^{-} > 1$ and thus we conclude that the sequence ${u_{n}}_{n \in N}$ is bounded. Therefore, we may suppose (for a subsequence if necessary) that
$\begin{aligned} u_{n} ⇀ u in W_{0}^{1, H_{\log}} (Ω) and u_{n} \to u in both L^{(q^{+} + η) θ} (Ω) and L^{l} (Ω) . \end{aligned}$
(3.10)
Now, from (3.3) and (3.7) taking $v = u_{n} - u$ we see that
$\begin{aligned} lim_{n \to + \infty} {K [\int_{Ω} (\frac{1}{p (x)} | \nabla u_{n} |^{p (x)} + \frac{μ (x)}{q (x)} | \nabla u_{n} |^{q (x)} \log (e + | \nabla u_{n} |)) d x] \\ \times \int_{Ω} A (u_{n}) \cdot \nabla (u_{n} - u) d x - \int_{Ω} f (x, u_{n}) (u_{n} - u) d x} = 0. \end{aligned}$
We remark that using hypothesis (H3)(i) with $ε = 1$ , the Hölder’s inequality and (3.10) we can derive that
$\begin{aligned} | \int_{Ω} f (x, u_{n}) (u_{n} - u) d x | & ⩽ \int_{Ω} ((q^{+} + η) θ | u_{n} |^{(q^{+} + η) θ - 1} + l δ_{1} | u_{n} |^{l - 1}) | u_{n} - u | d x \\ ⩽ (q^{+} + η) θ ‖ u_{n} ‖_{((q^{+} + η) θ)^{^{'}}}^{(q^{+} + η) θ} ‖ u_{n} - u ‖_{(q^{+} + η) θ} \\ + l δ_{1} ‖ u_{n} ‖_{l^{'}}^{l} ‖ u_{n} - u ‖_{l} \\ \to 0 as n \to + \infty . \end{aligned}$
Also, from Proposition 2.3 we know that the operator $A$ introduced in (2.1) is continuous and bounded. Consequently, according to (3.10) we have
$\begin{aligned} \int_{Ω} A (u_{n}) \cdot \nabla (u_{n} - u) d x \to 0 as n \to + \infty \end{aligned}$
which gives
$\begin{aligned} \underset{n \to + \infty}{lim sup} ⟨ A (u_{n}), u_{n} - u ⟩ ⩽ 0. \end{aligned}$
Now, taking into account that the operator $A$ is of $S_{+}$ -type (see again Proposition 2.3), $K (t) > 0$ for all $t ⩾ 1$ (see (3.9)) and as (3.6) holds, we conclude that
$\begin{aligned} u_{n} \to u in W_{0}^{1, H_{\log}} (Ω) . \end{aligned}$
Therefore, the functional Φ satisfies the Palais–Smale condition and the assertion is proved.

Step 4. Finally, as $Φ (0) = 0$ , Steps $1 - 3$ permit us to apply the mountain pass theorem which ensures the existence of a nontrivial critical value of Φ. Taking into account that the critical points of Φ are the weak solutions of problem (1.1) (see (3.1) and (3.3)), the proof is completed.

4. Multiplicity of solutions

The aim of this section is to provide a multiplicity result of solutions for problem (1.1). We establish such result thanks to a variational approach based on the use of Fountain theorem [25, Theorem 3.6] which is a variant of the symmetric mountain pass theorem. With this end, we here consider new assumptions on the nonlinearity f. More precisely, we replace hypothesis (H3)(i) with the following condition:

(H4)

f is odd with respect to the second variable and there exists $\bar{l} \in (p^{+}, (p^{*})^{-})$ such that

\begin{aligned} | f (x, t) | ⩽ \bar{c} (1 + | t |^{\bar{l} - 1}) \end{aligned}

for some

\bar{c} > 0

, for a.a.

x \in Ω

and for all

t \in R

Now, we recall that $W_{0}^{1, H_{\log}} (Ω)$ is a separable and reflexive Banach space and thus we can find sequences

\begin{aligned} {w_{n}}_{n \in N} \subset W_{0}^{1, H_{\log}} (Ω) and {h_{n}}_{n \in N} \subset [W_{0}^{1, H_{\log}} (Ω)]^{*} \end{aligned}

such that

\begin{aligned} W_{0}^{1, H_{\log}} (Ω) := \bar{span {w_{n} : n \in N}}, {[W_{0}^{1, H_{\log}} (Ω)]}^{*} := \bar{span {h_{n} : n \in N}} \\ and in addition ⟨ h_{j}, w_{n} ⟩ := {\begin{cases} 1 & if n = j, \\ 0 & if n \neq j . \end{cases} \end{aligned}

Then, we put

\begin{aligned} W_{n} := span {w_{n}}, {\tilde{W}}_{n} := \overset{n}{⨁_{j = 1}} W_{j}, {\hat{W}}_{n} := \overset{+ \infty}{⨁_{j = n}} {\tilde{W}}_{j} \end{aligned}

and

\begin{aligned} ω_{n} := sup_{u \in {\tilde{W}}_{n}, ‖ u ‖ = 1} ‖ u ‖_{\bar{l}}, \end{aligned}

with

\bar{l}

being as given in hypothesis (H4) (so as

\bar{l} < (p^{*})^{-}

we can embed

W_{0}^{1, H_{\log}} (Ω)

compactly in

L^{\bar{l}} (Ω)

Next, we can state and prove our multiplicity result. Theorem 4.1.

Let hypotheses (H1), (H2), (H3)(ii) and (H4) be satisfied. Then, problem (1.1) has infinitely many weak solutions in $W_{0}^{1, H_{\log}} (Ω)$ .

Proof.

In order to prove the claim, our strategy is to apply the Fountain theorem to the functional Φ defined in (3.2). To this purpose, we first note that as f is odd with respect to the second variable (see hypothesis (H4)) we have that Φ is an even functional. Also, reasoning as done in Step 3 of the proof of Theorem 3.2 we can see that Φ satisfies the Palais–Smale condition. Thus, in order to use the Fountain theorem we need only to show that for all $n ⩾ 1$ there are $σ_{n} > γ_{n} > 0$ such that

\begin{aligned} max {Φ (u) : u \in {\tilde{W}}_{n}, | | u | | = σ_{n}} ⩽ 0 \end{aligned}

(4.1)

and

\begin{aligned} Λ_{n} := inf {Φ (u) : u \in {\hat{W}}_{n}, | | u | | = γ_{n}} \to + \infty as n \to + \infty . \end{aligned}

(4.2)

We start by providing

γ_{n} > 0

so that (4.2) holds. We point out that from hypothesis (H4) we get that

\begin{aligned} | F (x, t) | ⩽ C (| t | + | t |^{\bar{l}}) \end{aligned}

(4.3)

for some

C > 0

, for a.a.

x \in Ω

and for all

t \in R

Also, due to hypothesis (H2)(ii) there exists $k > 0$ such that for all $u \in {\tilde{W}}_{n}$ with $‖ u ‖ ⩾ (q^{+})^{\frac{1}{p^{-}}} > 1$ it results

\begin{aligned} K [\int_{Ω} (\frac{1}{p (x)} | \nabla u |^{p (x)} + \frac{μ (x)}{q (x)} | \nabla u |^{q (x)} \log (e + | \nabla u |)) d x] ⩾ k \end{aligned}

(4.4)

(see (3.9) and (3.6)).

Thus, using hypothesis (H2)(i), (4.3), (4.4), Propisition 2.1(ii) and the Hölder inequality, for all $u \in {\hat{W}}_{n}$ with $‖ u ‖ ⩾ (q^{+})^{\frac{1}{p^{-}}} > 1$ we derive that

\begin{aligned} Φ (u) & ⩾ \frac{1}{q^{+} θ} K [\int_{Ω} (\frac{1}{p (x)} | \nabla u |^{p (x)} + \frac{μ (x)}{q (x)} | \nabla u |^{q (x)} \log (e + | \nabla u |)) d x] \\ \times \int_{Ω} (| \nabla u |^{p (x)} + μ (x) | \nabla u |^{q (x)} \log (e + | \nabla u |)) d x \\ - C \int_{Ω} | u | d x - C \int_{Ω} | u |^{\bar{l}} d x \\ ⩾ \frac{k}{q^{+} θ} ‖ u ‖^{p^{-}} - C | Ω |^{\frac{\bar{l} - 1}{\bar{l}}} ‖ u ‖_{\bar{l}} - C ‖ u ‖_{\bar{l}}^{\bar{l}} \\ ⩾ \frac{k}{q^{+} θ} ‖ u ‖^{p^{-}} - C ω_{n} | Ω |^{\frac{\bar{l} - 1}{\bar{l}}} ‖ u ‖ - C ω_{n}^{\bar{l}} ‖ u ‖^{\bar{l}} \\ ⩾ \frac{k}{q^{+} θ} ‖ u ‖^{p^{-}} - C ω_{n} | Ω |^{\frac{\bar{l} - 1}{\bar{l}}} ‖ u ‖^{\bar{l}} - C ω_{n}^{\bar{l}} ‖ u ‖^{\bar{l}} \\ = [\frac{k}{q^{+} θ} - C (ω_{n} | Ω |^{\frac{\bar{l} - 1}{\bar{l}}} + ω_{n}^{\bar{l}}) ‖ u ‖^{\bar{l} - p^{-}}] ‖ u ‖^{p^{-}} \end{aligned}

(4.5)

with the constant

C > 0

which may change from line to line. Now, we point out that from Lemma 7.1 of [16] we know that

ω_{n} \to 0

n \to + \infty

. Thus, if we choose

\begin{aligned} γ_{n} := {[\frac{k}{2 q^{+} θ} \frac{1}{C (ω_{n} | Ω |^{\frac{\bar{l} - 1}{\bar{l}}} + ω_{n}^{\bar{l}})}]}^{\frac{1}{\bar{l} - p^{-}}}, \end{aligned}

\bar{l} > p^{-}

due to hypothesis (H4), we have that

\begin{aligned} γ_{n} \to + \infty as n \to + \infty . \end{aligned}

(4.6)

Therefore, in accordance with (4.5), for all

u \in {\hat{W}}_{n}

with

‖ u ‖ = γ_{n}

and n large enough (which implies

γ_{n} > (q^{+})^{\frac{1}{p^{-}}}

due to (4.6)) we see that

\begin{aligned} Λ_{n} & ⩾ \frac{k}{2 q^{+} θ} γ_{n}^{p^{-}} \to + \infty as n \to + \infty, \end{aligned}

that is, (4.2) holds.

Next, we show that (4.1) is verified as well. To this purpose, we first note that as ${\tilde{W}}_{n}$ is finite dimensional, all the norms on ${\tilde{W}}_{n}$ are equivalent, see for example [20, Proposition 3.1.17, p.183]. Consequently, we can find a positive constant $c_{{\tilde{W}}_{n}}$ , independent of $u \in {\tilde{W}}_{n}$ , such that

\begin{aligned} c_{{\tilde{W}}_{n}} ‖ u ‖^{m} ⩽ ‖ u ‖_{m}^{m} \end{aligned}

being m as given in hypothesis (H3)(ii).

Now, using the previous inequality along with (3.4), (3.5) (where we take $ε = 1$ ) and (3.6), for $u \in {\tilde{W}}_{n}$ with $‖ u ‖ ⩾ (q^{+})^{\frac{1}{p^{-}}}$ we deduce that

\begin{aligned} Φ (u) & ⩽ K (1) {[\int_{Ω} (\frac{1}{p (x)} | \nabla u |^{p (x)} + \frac{μ (x)}{q (x)} | \nabla u |^{q (x)} \log (e + | \nabla u |)) d x]}^{θ} \\ - b_{1} \int_{Ω} | u |^{m} d x + b_{2} \int_{Ω} d x \\ ⩽ \frac{K (1)}{p^{-}} [ϱ_{H_{\log}} (\nabla u)]^{θ} - b_{1} ‖ u ‖_{m}^{m} + b_{2} | Ω | \\ ⩽ \frac{K (1)}{p^{-}} ‖ u ‖^{(q^{+} + η) θ} - b_{1} c_{{\tilde{W}}_{n}} ‖ u ‖^{m} + b_{2} | Ω | \end{aligned}

(we recall that

‖ u ‖ := ‖ \nabla u ‖_{H_{\log}} ⩾ (q^{+})^{\frac{1}{p^{-}}} > 1

gives

ϱ_{H_{\log}} (\nabla u) ⩽ ‖ u ‖^{q^{+} + η}

in accordance to Proposition 2.1(ii)). Taking into account that

m > (q^{+} + η) θ

by hypothesis (H3)(ii), it is sufficient to choose

σ_{n} > max {(q^{+})^{\frac{1}{p^{-}}}, γ_{n}}

large enough in order to infer that (4.1) holds.

According of all this, we can finally apply the Fountain theorem to functional Φ. In this way, we get the existence of an unbounded sequence of critical points of Φ. As such critical points are weak solutions for problem (1.1) (see (3.1) and (3.3)), the proof is concluded.

Footnotes

Funding

The author declares that no funds, grants, or other support were received during the preparation of this manuscript.

Competing interests

The author read and approved the final manuscript. The author has no relevant financial or non-financial interests to disclose.

Availability of data and material

This paper has no associated data and material.

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