Multiple solutions for nonlinear boundary value problems of Kirchhoff type on a double phase setting

Abstract

This paper deals with some classes of Kirchhoff type problems on a double phase setting and with nonlinear boundary conditions. Under general assumptions, we provide multiplicity results for such problems in the case when the perturbations exhibit a suitable behavior in the origin and at infinity, or when they do not necessarily satisfy the Ambrosetti–Rabinowitz condition. To this aim, we combine variational methods, truncation arguments and topological tools.

1 Introduction

In this paper, we study multiplicity results for some classes of double phase problems of Kirchhoff type with nonlinear boundary conditions. More precisely, we consider the problems

$$\begin{aligned} \left\{ \begin{aligned}&-M\left[ \displaystyle \int _\Omega \left( \frac{|\nabla u|^p}{p}+a(x)\frac{|\nabla u|^q}{q}\right) dx\right] \text{ div }\left( |\nabla u|^{p-2}\nabla u+a(x)|\nabla u|^{q-2}\nabla u\right) = h_1(x,u) \quad \text{ in } \Omega ,\\&M\left[ \displaystyle \int _\Omega \left( \frac{|\nabla u|^p}{p}+a(x)\frac{|\nabla u|^q}{q}\right) dx\right] \left( |\nabla u|^{p-2}\nabla u+a(x)|\nabla u|^{q-2}\nabla u\right) \cdot \nu = h_2(x,u) \quad \text{ on } \partial \Omega , \end{aligned}\right. \end{aligned}$$

(1.1)

and

$$\begin{aligned} \left\{ \begin{aligned}&-M\left( \int _{\Omega }|\nabla u|^p dx\right) \Delta _p u- M\left( \int _{\Omega }a(x) |\nabla u|^q dx \right) {{\,\textrm{div}\,}}\left( a(x) |\nabla u|^{q-2} \nabla u\right) = h_1(x, u) \quad \text {in } \Omega , \\&\left[ M\left( \int _{\Omega }|\nabla u|^p dx \right) |\nabla u|^{p-2}\nabla u+M\left( \int _{\Omega }a(x) |\nabla u|^q dx \right) a(x)|\nabla u|^{q-2}\nabla u\right] \cdot \nu = h_2(x,u) \quad \text{ on } \partial \Omega , \end{aligned}\right. \end{aligned}$$

(1.2)

where along the paper, and without further mentioning, $\Omega \subset {\mathbb {R}}^N$, $N>1$, is a bounded domain with $C^{1, \alpha }$- boundary $\partial \Omega$, $\alpha \in (0, 1]$, $\nu (x)$ is the outer unit normal of $\Omega$ at the point $x \in \partial \Omega$, $1<p<q<N$, and

$$\begin{aligned} q<p^*, \quad a:\overline{\Omega }\rightarrow [0, \infty ) \text { is in }L^\infty (\Omega ), \end{aligned}$$

(1.3)

with $p^*=Np/(N-p)$ being the critical Sobolev exponent, see (2.4) for its definition. We assume that $M:[0, \infty ) \rightarrow [0, \infty )$ is a continuous function satisfying the following assumptions:

$(M_1)$:

there exists $\theta \ge 1$ such that $t M(t)\le \theta {\mathscr {M}}(t)$ for all $t\in [0,\infty )$, where ${\mathscr {M}}(t) =\displaystyle \int _0^t M(\tau )\;d\tau$;

$(M_2)$:

for all $\tau >0$ there exists $\kappa =\kappa (\tau )>0$ such that $M(t)\ge \kappa$ for all $t\ge \tau$.

Moreover, $h_1:\Omega \times {\mathbb {R}}\rightarrow {\mathbb {R}}$ and $h_2:\partial \Omega \times {\mathbb {R}}\rightarrow {\mathbb {R}}$ are Carathéodory functions whose properties will be specified in the sequel, case by case. A classical model for M verifying $(M_1)$–$(M_2)$ due to Kirchhoff is given by $M(t)=a+bt^{\theta -1}$, where a, $b\ge 0$ and $a+b>0$.

Problems (1.1) and (1.2) are said 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 [28], where he introduced the functional

$$\begin{aligned} u\mapsto \int _\Omega \left( |\nabla u|^p+a(x)|\nabla u|^q\right) dx \quad \text{ with } 1<p<q,\quad a(\cdot )\ge 0, \end{aligned}$$

(1.4)

in order to provide models for strongly anisotropic materials. Indeed, the weight 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, (1.4) is a prototype of a functional whose integrands change their ellipticity according to the points where $a(\cdot )$ vanishes or not. In this direction, Zhikov found other mathematical applications for (1.4) in the study of duality theory and of the Lavrentiev gap phenomenon, as shown in [29,30,31]. Furthermore, (1.4) falls into the class of functionals with non-standard growth conditions, according to Marcellini’s definition given in [22, 23]. Following this line of research, Mingione et al. provide different regularity results for minimizers of (1.4), see [1, 2, 6, 7]. In [5], Colasuonno and Squassina analyze the eigenvalue problem with Dirichlet boundary condition of the double phase operator, explicitly appearing in (1.1), that is,

$$\begin{aligned} u\mapsto \text{ div }\left( |\nabla u|^{p-2}\nabla u+a(x)|\nabla u|^{q-2}\nabla u\right) , \end{aligned}$$

(1.5)

whose energy functional is given by (1.4).

Starting from [5], several authors studied existence and multiplicity results for nonlinear problems driven by (1.5), such as in [8, 10,11,12,13,14, 17, 18, 21, 26] with the help of different variational techniques. In particular, in [14] Fiscella and Pinamonti provide existence and multiplicity results for Kirchhoff double phase problems but with Dirichlet boundary condition. That is, in [14] we have

$$\begin{aligned} \left\{ \begin{array}{ll} -M\left[ \displaystyle \int _\Omega \left( \frac{|\nabla u|^p}{p}+a(x)\frac{|\nabla u|^q}{q}\right) dx\right] \text{ div }\left( |\nabla u|^{p-2}\nabla u+a(x)|\nabla u|^{q-2}\nabla u\right) =f(x,u) &{} \text{ in } \Omega ,\\ u=0 &{} \text{ in } \partial \Omega , \end{array}\right. \end{aligned}$$

(1.6)

and

$$\begin{aligned} \left\{ \begin{array}{ll} -M\left( \displaystyle \int _\Omega |\nabla u|^p dx\right) \Delta _p u-M\left( \displaystyle \int _\Omega a(x)|\nabla u|^q dx\right) \text{ div }\left( a(x)|\nabla u|^{q-2}\nabla u\right) =f(x,u) &{} \text{ in } \Omega ,\\ u=0 &{} \text{ in } \partial \Omega , \end{array}\right. \end{aligned}$$

where the nonlinear subcritical term f satisfies the classical Ambrosetti–Rabinowitz (AR) condition, crucial to prove the boundedness of the so-called Palais-Smale sequences.

The aim of the present paper is to provide different multiplicity results for some classes of (1.1) and (1.2), that will heavily depend on the properties of the involved nonlinearities. More specifically, in the first part of the work, we will study existence of at least two constant sign solutions (more precisely, one nonnegative and one nonpositive), under the assumption that the nonlinearities satisfy some strong conditions such as the superlinearity at $\pm \infty$. These results are inspired by the truncation arguments used in [10], where the authors studied existence of constant sign solutions in a double phase setting but without the Kirchhoff coefficient, namely, with $M \equiv 1$. However, the presence of the nonlocal Kirchhoff coefficient in (1.1) and (1.2) makes the comparison analysis more intriguing than [10]. For this, our multiplicity result will depend on the value M(0), beside the first eigenvalue of the Robin eigenvalue problem for the p-Laplacian, as required in [10]. Other results for existence of solutions with sign information in a double phase setting can be found, for example, in [15, 16, 25].

In the second part of the article we will study multiplicity of solutions in the case when the nonlinearities on the right-hand side do not necessarily satisfy the (AR) condition. In this direction, we can mention [17, 18] where they consider problems (1.1) and (1.6) with $M\equiv 1$, namely without the Kirchhoff coefficient. In [18], Ge et al. exploit the classical mountain pass theorem and the Krasnoselskii’s genus theory in order to prove existence and multiplicity results for (1.6), replacing the (AR) condition with

$$\begin{aligned} {\mathcal {F}}(x,t)\le {\mathcal {F}}(x,s)+C_0,\quad \text{ where } {\mathcal {F}}(x,t) =tf(x,t)-qF(x,t) \text{ and } C_0>0 \nonumber \\ \end{aligned}$$

(1.7)

for a.e. $x\in \Omega$, $0<t<s$ or $s<t<0$, where F is the primitive of f with respect to the second variable. On the other hand, in [17] Gasiński and Winkert prove the existence of two solutions of (1.1) with $h_1(x,t)=f(x,t)-|t|^{p-2}t-a(x)|t|^{q-2}t$, with the following quasi-monotonic assumption for f

$$\begin{aligned} t\mapsto tf(x,t)-qF(x,t) \text{ is } \text{ nondecreasing } \text{ in } {\mathbb {R}}^+ \text{ and } \text{ nonincreasing } \text{ in } {\mathbb {R}}^-, \end{aligned}$$

(1.8)

for a.e. $x\in \Omega$. Because of the presence of a nonlocal Kirchhoff coefficient M in (1.1), similar hypothesis to (1.7) and (1.8) can not work, even if we assume a monotonic assumption for $t\mapsto \theta {\mathscr {M}}(t)-M(t)t$. This is a consequence of the fact that M in (1.1) does not depend on the norm of $W^{1,{\mathcal {H}}}(\Omega )$, that is the functional space where we look for the solutions to (1.1) and (1.2). Indeed, as shown in detail in Sect. 2, $W^{1,{\mathcal {H}}}(\Omega )$ is a suitable Musielak-Orlicz Sobolev space endowed with a norm of Luxemburg type. For this reason, we need different conditions for the nonlinearities in (1.1) and (1.2) than in [17, 18].

The paper is organized as follows. In Sect. 2, we recall the main properties of Musielak-Orlicz Sobolev spaces $W^{1,{\mathcal {H}}}(\Omega )$ and we state some technical lemmas concerning the Kirchhoff coefficient M. In Sect. 3, for each problem we prove the existence of two constant sign solutions, more precisely one nonnegative and one nonpositive, combining the ideas of [10, 14]. In order to show the results, we require some strong conditions on the nonlinearities, like the superlinearity at $\pm \infty$. We also point out that the proofs depend on the first eigenvalue of the Robin eigenvalue problem for the p-Laplacian. In the last part of the paper, namely Sect. 4, we show the existence of infinitely many solutions to a class of (1.1) and (1.2). In particular, we introduce suitable assumptions for the nonlinearities in order to avoid the (AR) condition.

2 Preliminaries

In this section we introduce the basic notation of our paper, the functional space where we find the solutions to (1.1), (1.2) and we present some technical lemmas for M that will be used in the sequel.

For all $1 \le r< \infty$ we denote by $L^r(\Omega )$ and $L^r(\Omega ; {\mathbb {R}}^{N})$ the usual Lebesgue spaces equipped with the norm $\Vert \cdot \Vert _r$. The corresponding Sobolev space is denoted by $W^{1,r}(\Omega )$, endowed with the norm $\Vert \cdot \Vert _{1,r}$. On the boundary $\partial \Omega$ of $\Omega$ we consider the $(N-1)$-dimensional Hausdorff measure $\sigma$ and denote by $L^r(\partial \Omega )$ the boundary Lebesgue space with corresponding norm $\Vert \cdot \Vert _{r, \partial \Omega }$. As matter of notations, for all $t\in {\mathbb {R}}$ we set $t^{\pm }:= \max \{\pm t, 0\}$ and similarly we denote by $u^{\pm }(\cdot ):= u(\cdot )^{\pm }$ the positive and negative parts of a function u.

The function ${\mathcal {H}}:\Omega \times [0,\infty )\rightarrow [0,\infty )$ defined as

$$\begin{aligned} {\mathcal {H}}(x,t):=t^p+a(x)t^q \quad \text{ for } \text{ a.e. } x \in \Omega \text{ and } \text{ for } \text{ all } t\in [0,\infty ), \end{aligned}$$

with $1<p<q$ and $0\le a \in L^1(\Omega )$, is a generalized N-function (N stands for nice), according to the definition in [9, 24], and satisfies the so-called $(\Delta _2)$ condition, that is,

$$\begin{aligned} {\mathcal {H}}(x,2t)\le 2^q{\mathcal {H}}(x,t) \quad \text{ for } \text{ a.e. } x \in \Omega \text{ and } \text{ for } \text{ all } t\in [0,\infty ). \end{aligned}$$

Then, we introduce the ${\mathcal {H}}$-modular function $\varrho _{{\mathcal {H}}}$ given by

$$\begin{aligned} \varrho _{{\mathcal {H}}}(u):=\int _\Omega {\mathcal {H}}(x,|u|) \;dx=\int _\Omega \left( |u|^p+a(x)|u|^q\right) dx. \end{aligned}$$

(2.1)

Therefore, by [24] we can define the Musielak-Orlicz space $L^{{\mathcal {H}}}(\Omega )$ as

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

endowed with the Luxemburg norm

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

From [5, 9], the space $L^{{\mathcal {H}}}(\Omega )$ is a separable, uniformly convex, Banach space. On the other hand, from [21, Proposition 2.1] we have the following relation between the norm $\Vert \cdot \Vert _{{\mathcal {H}}}$ and the ${\mathcal {H}}$-modular function.

Proposition 2.1

Assume that $u\in L^{{\mathcal {H}}}(\Omega )$, $(u_j)_j\subset L^{{\mathcal {H}}}(\Omega )$, and $c>0$. Then,

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

• (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) $\lim \limits _{j\rightarrow \infty }\Vert u_j\Vert _{{\mathcal {H}}} =0\,(\text {resp. } \infty )$ if and only if $\lim \limits _{j\rightarrow \infty } \varrho _{{\mathcal {H}}}(u_j)=0\,(\text {resp. } \infty )$.

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

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

endowed with the norm

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

(2.2)

Along the paper, we write $\Vert \nabla u\Vert _{{\mathcal {H}}}:=\Vert |\nabla u|\Vert _{{\mathcal {H}}}$ and $\varrho _{{\mathcal {H}}}(\nabla u):=\varrho _{{\mathcal {H}}}(|\nabla u|)$ in order to simplify the notation.

We also define the complete ${\mathcal {H}}$-modular function $\widehat{\varrho }_{{\mathcal {H}}}$ as

$$\begin{aligned} \widehat{\varrho }_{{\mathcal {H}}}(u):=\varrho _{{\mathcal {H}}}(\nabla u) +\varrho _{{\mathcal {H}}}(u) =\int _\Omega \left( |\nabla u|^p+a(x)| \nabla u|^q+|u|^p+a(x)|u|^q\right) dx. \end{aligned}$$

(2.3)

Hence, following the proof of [21, Proposition 2.1] we can relate $\widehat{\varrho }_{{\mathcal {H}}}$ and the norm $\Vert \cdot \Vert _{1, {\mathcal {H}}}$ as follows.

Proposition 2.2

Assume that $u\in W^{1,{\mathcal {H}}}(\Omega )$, $(u_j)_j\subset W^{1,{\mathcal {H}}}(\Omega )$, and $c>0$. Then,

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

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

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

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

• (v) $\lim \limits _{j\rightarrow \infty }\Vert u_j\Vert _{1,{\mathcal {H}}} =0\,(\text {resp. } \infty )$ if and only if $\lim \limits _{j\rightarrow \infty } \widehat{\varrho }_{{\mathcal {H}}}(u_j)=0\,(\text {resp. } \infty )$.

For all $1< p< N$ let $p^*$ and $p_*$ denote the critical Sobolev exponents of p, defined as

$$\begin{aligned} p^*= \displaystyle \frac{Np}{N-p} \quad \text {as well as} \quad p_*= \displaystyle \frac{(N-1)p}{N-p}. \end{aligned}$$

(2.4)

Furthermore, we define the weighted space

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

equipped with the seminorm

$$\begin{aligned} \Vert u\Vert _{q,a}:=\left( \int _\Omega a(x)|u|^q \;dx\right) ^{1/q}. \end{aligned}$$

In a similar way we can define $L^q_a(\Omega ; {\mathbb {R}}^{N})$ and the associated seminorm. Thus, by [10, Proposition 2.1] we have the following embeddings for $L^{\mathcal {H}}(\Omega )$ and $W^{1,\mathcal {H}}(\Omega )$.

Proposition 2.3

Let (1.3) be satisfied. Then, the following embeddings hold:

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

• (ii) $W^{1,\mathcal {H}}(\Omega )\hookrightarrow L^{\nu _1}(\Omega )$ is compact for all $\nu _1\in [1,p^*)$;

• (iii) $W^{1,\mathcal {H}}(\Omega )\hookrightarrow L^{\nu _2}(\partial \Omega )$ is compact for all $\nu _2\in [1,p_*)$;

• (iv) $L^q(\Omega )\hookrightarrow L^{\mathcal {H}}(\Omega )\hookrightarrow L_a^q(\Omega )$ are continuous;

• (v) $W^{1,\mathcal {H}}(\Omega )\hookrightarrow L^{\mathcal {H}}(\Omega )$ is compact.

Let us define the operator $L:W^{1,{\mathcal {H}}}(\Omega ) \rightarrow \left( W^{1,{\mathcal {H}}}(\Omega )\right) ^*$ such that

$$\begin{aligned} \langle L(u),v\rangle _{{\mathcal {H}}}:=\int _\Omega \left( |\nabla u|^{p-2}+a(x)|\nabla u|^{q-2}\right) \nabla u\cdot \nabla v \;dx, \end{aligned}$$

(2.5)

for all u, $v\in W^{1,{\mathcal {H}}}(\Omega )$. Here, $\left( W^{1,{\mathcal {H}}}(\Omega )\right) ^*$ denotes the dual space of $W^{1,{\mathcal {H}}}(\Omega )$ and $\langle \cdot \,,\cdot \rangle _{{\mathcal {H}}}$ is the related dual pairing. Then we have the following result, see [21, Proposition 3.1-(ii)].

Proposition 2.4

The mapping $L:W^{1,{\mathcal {H}}}(\Omega )\rightarrow \left( W^{1,{\mathcal {H}}} (\Omega )\right) ^*$ is of $(S_+)$ type, that is, if $u_j\rightharpoonup u$ in $W^{1,{\mathcal {H}}}(\Omega )$ and $\limsup \limits _{j\rightarrow \infty }\langle L(u_j)-L(u), u_j-u\rangle \le 0$, then $u_j\rightarrow u$ in $W^{1,{\mathcal {H}}}(\Omega )$.

Now, we introduce some technical results related to the Kirchhoff coefficient M, under the assumptions that $(M_1)$–$(M_2)$ hold true. To this aim, we observe that integrating $(M_1)$ on (1, t), when $t\ge 1$, gives

$$\begin{aligned} {\mathscr {M}}(t)\le {\mathscr {M}}(1) t^\theta \qquad \text{ for } \text{ all } t\ge 1. \end{aligned}$$

(2.6)

Moreover, we set

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

(2.7)

Thus, we have the following lemmas.

Lemma 2.5

Let $u\in W^{1,\mathcal {H}}(\Omega )$ be such that $\left\| {u}\right\| _{1,{\mathcal {H}}}>1$. Then, there exist ${A_1}$, ${A_2}>0$ such that

$$\begin{aligned} \varrho _{{\mathcal {H}}}(\nabla u) M[\phi _{{\mathcal {H}}}(\nabla u)] +\varrho _{{\mathcal {H}}}(u) \ge {A_1}\Vert u\Vert _{1, {\mathcal {H}}}^p, \end{aligned}$$

(2.8)

and

$$\begin{aligned} M(\Vert \nabla u\Vert _p^p)\Vert \nabla u\Vert _p^p+M(\Vert \nabla u\Vert _{q,a}^q)\Vert \nabla u\Vert _{q,a}^q+\varrho _{{\mathcal {H}}}(u) \ge {A_2}\Vert u\Vert _{1,{\mathcal {H}}}^p. \end{aligned}$$

(2.9)

Proof

We first prove (2.8). Since $\Vert u\Vert _{1, {\mathcal {H}}} > 1$, by Proposition 2.2-(ii) and (iv) we have that

$$\begin{aligned} \widehat{\varrho }_{{\mathcal {H}}}(u)\ge \Vert u\Vert _{1, {\mathcal {H}}}^p>1. \end{aligned}$$

This in particular implies that

$$\begin{aligned} \Vert u\Vert _{1,{\mathcal {H}}}^p\le {\left\{ \begin{array}{ll} \varrho _{{\mathcal {H}}}(\nabla u)+\varrho _{{\mathcal {H}}}(u) &{}\text{ if }\ \varrho _{{\mathcal {H}}}(\nabla u)\ge 1/2,\\ \varrho _{{\mathcal {H}}}(\nabla u)+\varrho _{{\mathcal {H}}}(u) \le 2\varrho _{{\mathcal {H}}}( u) &{}\text{ if }\ \varrho _{{\mathcal {H}}}(\nabla u) <1/2\ \text{ and }\ \varrho _{{\mathcal {H}}}(u)\ge 1/2. \end{array}\right. } \end{aligned}$$

(2.10)

First suppose that $\varrho _{{\mathcal {H}}}(\nabla u) \ge 1/2$. Then it follows that

$$\begin{aligned} \phi _{{\mathcal {H}}}(\nabla u) \ge \frac{1}{q} \varrho _{{\mathcal {H}}}(\nabla u) \ge \frac{1}{2q}. \end{aligned}$$

Therefore we can use hypothesis ($M_2$) with $\displaystyle \tau = \frac{1}{2q}$ to find $\displaystyle \kappa = \kappa \left( \frac{1}{2q}\right) >0$ such that

$$\begin{aligned} M(t) \ge \kappa \quad \text {for all } t \ge \frac{1}{2q}. \end{aligned}$$

(2.11)

Then equation (2.11) and the first line of equation (2.10) give

$$\begin{aligned} \varrho _{{\mathcal {H}}}(\nabla u) M[\phi _{{\mathcal {H}}}(\nabla u)]+ \varrho _{{\mathcal {H}}}(u)&\ge \kappa \varrho _{{\mathcal {H}}}(\nabla u)+ \varrho _{{\mathcal {H}}}(u) \nonumber \\&\ge \min \{1, \kappa \} \widehat{\varrho }_{{\mathcal {H}}}(u) \nonumber \\&\ge \min \{1, \kappa \} \Vert u\Vert _{1, {\mathcal {H}}}^p. \end{aligned}$$

(2.12)

Suppose now that $\varrho _{{\mathcal {H}}}(\nabla u)< 1/2$ and $\varrho _{{\mathcal {H}}}(u) \ge 1/2$. Then the nonnegativity of M and the second line of (2.10) give

$$\begin{aligned} \varrho _{{\mathcal {H}}}(\nabla u) M[\phi _{{\mathcal {H}}}(\nabla u)]+ \varrho _{{\mathcal {H}}}(u) \ge \frac{1}{2} \Vert u\Vert _{1, {\mathcal {H}}}^p. \end{aligned}$$

(2.13)

Combining (2.12) and (2.13) gives (2.8).

Now, we prove (2.9). When $\varrho _{{\mathcal {H}}}(\nabla u)<1/2$, by (2.10) we obtain

$$\begin{aligned} M(\Vert \nabla u\Vert _p^p)\Vert \nabla u\Vert _p^p+M(\Vert \nabla u\Vert _{q,a}^q) \Vert \nabla u\Vert _{q,a}^q+\varrho _{{\mathcal {H}}}(u) \ge \frac{1}{2}\Vert u\Vert _{1,{\mathcal {H}}}^p. \end{aligned}$$

(2.14)

On the other hand, if $\varrho _{{\mathcal {H}}}(\nabla u)\ge 1/2$ we distinguish among three situations: either $\Vert \nabla u\Vert ^p_p\ge 1/4$ and $\Vert \nabla u\Vert ^q_{q,a}\ge 1/4$; or $\Vert \nabla u\Vert ^p_p\ge 1/4$ and $\Vert \nabla u\Vert ^q_{q,a}<1/4$, which yields $2\Vert \nabla u\Vert _p^p\ge \Vert \nabla u\Vert _p^p+\Vert \nabla u\Vert _{q,a}^q$; or $\Vert \nabla u\Vert ^p_p<1/4$ and $\Vert \nabla u\Vert ^q_{q,a}\ge 1/4$, which yields $2\Vert \nabla u\Vert _{q,a}^q\ge \Vert \nabla u\Vert _p^p+\Vert \nabla u\Vert _{q,a}^q$. Thus, by $(M_2)$ and Proposition 2.1, we get

$$\begin{aligned}&M(\Vert \nabla u\Vert _p^p)\Vert \nabla u\Vert _p^p +M(\Vert \nabla u\Vert _{q,a}^q)\Vert \nabla u\Vert _{q,a}^q\nonumber \\&\quad \ge {\left\{ \begin{array}{ll} \kappa \varrho _{{\mathcal {H}}}(\nabla u), &{}\text{ if }\ \Vert \nabla u\Vert ^p_p\ge 1/4\ \text{ and }\ \Vert \nabla u\Vert ^q_{q,a}\ge 1/4,\\ \frac{\kappa }{2}\varrho _{{\mathcal {H}}}(\nabla u), &{}\text{ if }\ \Vert \nabla u\Vert ^p_p\ge 1/4\ \text{ and }\ \Vert \nabla u\Vert ^q_{q,a}<1/4,\\ \frac{\kappa }{2}\varrho _{{\mathcal {H}}}(\nabla u), &{}\text{ if }\ \Vert \nabla u\Vert ^p_p<1/4\ \text{ and }\ \Vert \nabla u\Vert ^q_{q,a}\ge 1/4, \end{array}\right. } \end{aligned}$$

(2.15)

with $\kappa =\kappa (1/4)>0$ given in $(M_2)$ with $\tau =1/4$. Combining (2.14) and (2.15) gives (2.9). $\square$

Lemma 2.6

Let $u\in W^{1,\mathcal {H}}(\Omega )$ be such that $\left\| {u}\right\| _{1,{\mathcal {H}}}>1$. Then, there exist ${B_1}$, ${B_2}>0$ such that

$$\begin{aligned} {{\mathscr {M}}}[\phi _{{\mathcal {H}}}(\nabla u)]+ \varrho _{{\mathcal {H}}}(u) \le {B_1}(1+\Vert u\Vert _{1, {\mathcal {H}}}^q+\Vert u\Vert _{1, {\mathcal {H}}}^{q\theta }) \end{aligned}$$

(2.16)

and

$$\begin{aligned} {{\mathscr {M}}}({\left\| \nabla u\right\| }_{p}^p)+{{\mathscr {M}}}({\left\| \nabla u\right\| }_{q,a}^q) +\varrho _{{\mathcal {H}}}(u) \le {B_2}(1+\Vert u\Vert _{1, {\mathcal {H}}}^q+\Vert u\Vert _{1, {\mathcal {H}}}^{q\theta }). \end{aligned}$$

(2.17)

Proof

Let us fix $u\in W^{1,\mathcal {H}}(\Omega )$ such that $\left\| {u}\right\| _{1,{\mathcal {H}}}>1$. We first prove (2.16).

If $\varrho _{{\mathcal {H}}}(\nabla u)\ge q$, then $\phi _{\mathcal {H}}(\nabla u)\ge \frac{1}{q}\varrho _{\mathcal {H}}(\nabla u)\ge 1$, and so by (2.6) and Proposition 2.1 we get

$$\begin{aligned} {\mathscr {M}}[\phi _{{\mathcal {H}}}(\nabla u)] \le \displaystyle \frac{{\mathscr {M}}(1)}{p^\theta }[\varrho _{{\mathcal {H}}} (\nabla u)]^{\theta }\le \displaystyle \frac{{\mathscr {M}}(1)}{p^\theta } \Vert \nabla u\Vert _{{\mathcal {H}}}^{q\theta }. \end{aligned}$$

(2.18)

If conversely $\varrho _{{\mathcal {H}}}(\nabla u)\le q$, then $\phi _{\mathcal {H}}(\nabla u)\le \varrho _{{\mathcal {H}}}(\nabla u)\le q$. Therefore

$$\begin{aligned} {\mathscr {M}}[\phi _{{\mathcal {H}}}(\nabla u)]\le \displaystyle {{\mathcal {M}}}, \end{aligned}$$

(2.19)

where $\displaystyle {\mathcal {M}}=\max _{t\in [0,q]} {\mathscr {M}}(t)\in (0,\infty )$ by $(M_2)$ and the continuity of M. Combining (2.18), (2.19) with Proposition 2.1 we obtain

$$\begin{aligned} {\mathscr {M}}[\phi _{{\mathcal {H}}}(\nabla u)]+\varrho _{{\mathcal {H}}}(u)&\le {{\mathcal {M}}}+\displaystyle \frac{{\mathscr {M}}(1)}{p^\theta } \Vert \nabla u\Vert _{{\mathcal {H}}}^{q\theta }+\varrho _{{\mathcal {H}}}(u)\\&\le {{\mathcal {M}}}+\displaystyle {\mathscr {M}}(1) \Vert u\Vert _{1,{\mathcal {H}}}^{q\theta }+\Vert u\Vert _{1,{\mathcal {H}}}^{q}. \end{aligned}$$

In order to prove (2.17) we observe that, since $\left\| {u}\right\| _{1,{\mathcal {H}}}>1$, from Proposition 2.2 we have that $\widehat{\varrho }_{\mathcal {H}}(u)\le \left\| {u}\right\| _{1,{\mathcal {H}}}^q$. We distinguish among four cases. If $\left\| \nabla u\right\| _p\ge 1$ and $\left\| \nabla u\right\| _{q,a}\ge 1$, then

$$\begin{aligned} {{\mathscr {M}}}({\left\| \nabla u\right\| }_{p}^p)+{{\mathscr {M}}}({\left\| \nabla u\right\| }_{q,a}^q)+\varrho _{{\mathcal {H}}}(u)&\le {{\mathscr {M}}}(1){\left\| \nabla u\right\| }_{p}^{p\theta }+{{\mathscr {M}}} (1){\left\| \nabla u\right\| }_{q,a}^{q\theta }+\varrho _{{\mathcal {H}}}(u)\nonumber \\&\le 2{{\mathscr {M}}}(1)\widehat{\varrho }_{{\mathcal {H}}}(u)^\theta +\widehat{\varrho }_{{\mathcal {H}}}(u)\nonumber \\&\le \max \{1,2{{\mathscr {M}}}(1)\}(\left\| {u}\right\| _{1,{\mathcal {H}}}^{q\theta }+\left\| {u}\right\| _{1,{\mathcal {H}}}^q). \end{aligned}$$

(2.20)

If ${\left\| \nabla u\right\| }_{p}<1$ and ${\left\| \nabla u\right\| }_{q,a}\ge 1$ it holds that

$$\begin{aligned} {{\mathscr {M}}}({\left\| \nabla u\right\| }_{p}^p)+{{\mathscr {M}}}({\left\| \nabla u\right\| }_{q,a}^q)+\varrho _{{\mathcal {H}}}(u)&\le \widetilde{{{\mathcal {M}}}}+{{\mathscr {M}}}(1) {\left\| \nabla u\right\| }_{q,a}^{q\theta }+\varrho _{{\mathcal {H}}}(u)\nonumber \\&\le \widetilde{{{\mathcal {M}}}}+{{\mathscr {M}}}(1)\widehat{\varrho }_{{\mathcal {H}}}(u)^\theta +\widehat{\varrho }_{{\mathcal {H}}}(u)\nonumber \\&\le \max \{1,\widetilde{{{\mathcal {M}}}},{{\mathscr {M}}}(1)\} (1+\left\| {u}\right\| _{1,{\mathcal {H}}}^{q\theta }+\left\| {u}\right\| _{1,{\mathcal {H}}}^q), \end{aligned}$$

(2.21)

with $\displaystyle \widetilde{{\mathcal {M}}} =\max _{t\in [0,1]}{\mathscr {M}}(t)$. If ${\left\| \nabla u\right\| }_{p}\ge 1$ and ${\left\| \nabla u\right\| }_{q,a}<1$ the estimate is the same as above. Finally, when ${\left\| \nabla u\right\| }_{p}<1$ and ${\left\| \nabla u\right\| }_{q,a}<1$, it holds that

$$\begin{aligned} {{\mathscr {M}}}({\left\| \nabla u\right\| }_{p}^p)+{{\mathscr {M}}}({\left\| \nabla u\right\| }_{q,a}^q)+\varrho _{{\mathcal {H}}}(u)\le 2 \widetilde{{{\mathcal {M}}}}+\widehat{\varrho }_{{\mathcal {H}}}(u)\le \max \{2 \widetilde{{{\mathcal {M}}}},1\}(1+\left\| {u}\right\| _{1,{\mathcal {H}}}^q).\nonumber \\ \end{aligned}$$

(2.22)

Combining (2.20)-(2.22) we get (2.17). $\square$

We now recall some basic properties on the spectrum of the negative p-Laplacian with Robin boundary condition. We refer to the paper of L$\hat{\text {e}}$ [19] for further details. The p-Laplacian eigenvalue problem with Robin boundary condition is given by

$$\begin{aligned} \left\{ \begin{aligned}&-\Delta _p u= \lambda |u|^{p-2} u \quad \quad \text {in } \Omega , \\&|\nabla u|^{p-2} \nabla u \cdot \nu = -\beta |u|^{p-2} u \quad \text {on } \partial \Omega , \end{aligned}\right. \end{aligned}$$

(2.23)

with $\beta >0$. It is well known that there exists a smallest eigenvalue $\lambda _{1,p}>0$ of (2.23) which is isolated, simple, and can be variationally characterized by

$$\begin{aligned} \lambda _{1,p}= \inf _{u \in W^{1,p}(\Omega ), u \ne 0} \frac{\displaystyle \int _{\Omega }|\nabla u|^p dx+ \beta \int _{\partial \Omega }|u|^p d\sigma }{\displaystyle \int _{\Omega }|u|^p dx}. \end{aligned}$$

(2.24)

Moreover, let $u_{1,p}$ be the normalized (that is, $\Vert u_{1,p}\Vert _p= 1$), positive eigenfunction corresponding to $\lambda _{1,p}$. It is known that $u_{1,p} \in \text {int} \left( C^1(\overline{\Omega })_+\right)$, where

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

We conclude this section with a result which will enable us to obtain the existence of infinitely many solutions to (1.1) and (1.2). To this aim, let X be a Banach space. We recall that a functional $E:X \rightarrow {\mathbb {R}}$ satisfies the Cerami condition (C) if every sequence $(u_j)_j \subset X$ such that

$$\begin{aligned} (E(u_j))_j \text{ is } \text{ bounded } \text{ and } (1+\Vert u_j\Vert _{1,{\mathcal {H}}})E' (u_j)\rightarrow 0 \text{ in } X^{*} \text{ as } j\rightarrow \infty \end{aligned}$$

(2.25)

admits a convergent subsequence in X; see for instance [3]. We say that $(u_j)_j$ is a Cerami sequence for E if it satisfies (2.25).

Let us now suppose that X is a reflexive and separable Banach space. It is well known that there exist $(e_j)_j \subset X$ and $(e^*_j)_j \subset X^*$ such that

$$\begin{aligned} X= \overline{\textrm{span}\{e_j:\, j\in {{\mathbb {N}}}\}} \quad \text {as well as} \quad X^*=\overline{\textrm{span} \{e_j^*:\, j\in {{\mathbb {N}}}\}} \end{aligned}$$

(2.26)

and

$$\begin{aligned} \langle e_i^*, e_j\rangle ={\left\{ \begin{array}{ll} 1 &{} \text {if } i=j, \\ 0 &{} \text {if } i\ne j. \end{array}\right. } \end{aligned}$$

(2.27)

For all $j\in {\mathbb {N}}$ we set

$$\begin{aligned} X_j:=\textrm{span}\{e_j\}, \qquad Y_j:=\bigoplus _{i=1}^j X_i, \qquad Z_j:=\bigoplus _{i=j}^{\infty } X_i. \end{aligned}$$

(2.28)

We can then state the following result, given in [20, Theorem 2.9], which is a variant of the classical Fountain theorem [27, Theorem 3.6] for functionals that satisfy the Cerami condition instead of the Palais-Smale condition.

Theorem 2.7

Let $E \in C^1(X, {\mathbb {R}})$ satisfy the Cerami condition (C) and be such that $E(-u)= E(u)$. Moreover, suppose that for every $j \in {\mathbb {N}}$ there exist $\rho _j> \gamma _j> 0$ such that

• (i) $\displaystyle b_j:= \inf _{u \in Z_j, \Vert u\Vert = \gamma _j} E(u) \rightarrow \infty$ as $j \rightarrow \infty$,

• (ii) $\displaystyle a_j:= \max _{u \in Y_j, \Vert u\Vert = \rho _j} E(u) \le 0$.

Then, E has a sequence of critical points $(u_j)_j$ such that $E(u_j) \rightarrow \infty$.

3 Constant sign solutions

In this section we prove the existence of constant-sign solutions to a class of (1.1) and (1.2) with superlinear nonlinearities. More specifically, we consider

$$\begin{aligned} h_1(x,t)&= (\zeta -\vartheta ) |t|^{p-2}t- a(x) |t|^{q-2}t- f(x,t) \quad \text {for a.e. } x \in \Omega , \nonumber \\ h_2(x,t)&= -\beta |t|^{p-2}t \quad \text {for a.e. } x \in \partial \Omega , \end{aligned}$$

(3.1)

for all $t \in {\mathbb {R}}$, where $\beta >0$, $\zeta>\vartheta >0$ are parameters to be further specified, and f is a Carathéodory function that satisfies suitable structure conditions stated below. Then problem (1.1) can be rewritten as

$$\begin{aligned} \left\{ \begin{aligned}&-M\left[ \displaystyle \int _\Omega \left( \frac{|\nabla u|^p}{p}+a(x)\frac{|\nabla u|^q}{q}\right) dx\right] \text{ div }\left( |\nabla u|^{p-2}\nabla u+a(x)|\nabla u|^{q-2}\nabla u\right) \\&\qquad = (\zeta - \vartheta ) |u|^{p-2} u- a(x) |u|^{q-2} u- f(x, u) \quad \text{ in } \Omega ,\\&M\left[ \displaystyle \int _\Omega \left( \frac{|\nabla u|^p}{p}+a(x)\frac{|\nabla u|^q}{q}\right) dx\right] \left( |\nabla u|^{p-2}\nabla u+a(x)|\nabla u|^{q-2}\nabla u\right) \cdot \nu = -\beta |u|^{p-2} u \quad \text{ on } \partial \Omega . \end{aligned}\right. \end{aligned}$$

(3.2)

We assume that the nonlinearity $f:\Omega \times {\mathbb {R}}\rightarrow {\mathbb {R}}$ is a Carathéodory function such that

($f_1$):

f is bounded on bounded subsets of $\Omega \times {\mathbb {R}}$;

($f_2$):

It holds

$$\begin{aligned} \lim _{t \rightarrow \pm \infty } \frac{f(x,t)}{ |t|^{q-2} t}= \infty \quad \text {uniformly for a.e. } x \in \Omega ; \end{aligned}$$

($f_3$):

It holds

$$\begin{aligned} \lim _{t \rightarrow 0} \frac{f(x, t)}{|t|^{p-2}t}= 0 \quad \text {uniformly for a.e. } x \in \Omega . \end{aligned}$$

A classical model for f satisfying $(f_1)$–$(f_3)$ is given by $f(x,t):=w(x)|t|^{k-2}t$, where $k>q$ and $w\in L^{\infty }(\Omega )$ with $\inf _\Omega w>0$. We say that $u \in W^{1, {\mathcal {H}}}(\Omega )$ is a weak solution to (3.2) if it satisfies

$$\begin{aligned}&M\left[ \phi _{{\mathcal {H}}}(\nabla u)\right] \int _{\Omega }\left( |\nabla u|^{p-2} \nabla u+ a(x) |\nabla u|^{q-2} \nabla u \right) \cdot \nabla \varphi \;dx\\&\qquad + \int _{\Omega }\left( \vartheta |u|^{p-2}u+ a(x) |u|^{q-2}u \right) \varphi \;dx \\&\quad = \int _{\Omega }\left( \zeta |u|^{p-2} u-f(x, u)\right) \varphi \;dx- \beta \int _{\partial \Omega }|u|^{p-2} u \varphi \;d\sigma \end{aligned}$$

for all $\varphi \in W^{1, {\mathcal {H}}}(\Omega )$.

Our existence result for problem (3.2) reads as follows.

Theorem 3.1

Let (1.3), $(M_1)$–$(M_2)$, and $(f_1)$–$(f_3)$ hold true. Moreover, assume that $\vartheta > 0$ and

$$\begin{aligned} \zeta > \vartheta + \max \{1,M(0)\}\lambda _{1, p}, \end{aligned}$$

(3.3)

where $\lambda _{1, p}$ is the first eigenvalue of the Robin eigenvalue problem given in (2.24). Then, there exist two nontrivial weak solutions $\widetilde{u}$, $\underline{u} \in W^{1, {\mathcal {H}}}(\Omega ) \cap L^{\infty }(\Omega )$ to problem (3.2), such that $\widetilde{u} \ge 0$ and $\underline{u} \le 0$.

Proof

First of all we observe that hypothesis $(f_2)$ allows us to find a constant $A= A(\zeta )>1$ such that

$$\begin{aligned} f(x,t)t \ge \zeta |t|^q \quad \text {for a.e. } x \in \Omega \text { and all } |t| \ge A. \end{aligned}$$

(3.4)

We start with the existence of a nonnegative solution of (3.2). From (3.4) we can take a constant function $u_0 \in (A, \infty )$ and use the fact that $p< q$ to achieve

$$\begin{aligned} \zeta u_0^{p-1}- f(x, u_0) \le 0 \quad \text {for a.e. } x \in \Omega . \end{aligned}$$

(3.5)

Moreover we consider the cut-off functions $b^{+}:\Omega \times {\mathbb {R}}\rightarrow {\mathbb {R}}$ and $b_{\beta }^{+}:\partial \Omega \times {\mathbb {R}}\rightarrow {\mathbb {R}}$ given by

$$\begin{aligned}&b^+(x, t)={\left\{ \begin{array}{ll} 0 &{} \text {if } t< 0 \\ \zeta t^{p-1}- f(x, t) &{} \text {if } 0 \le t< u_0 \\ \zeta u_0^{p-1}- f(x, u_0) &{} \text {if } t \ge u_0 \end{array}\right. },\nonumber \\&b_{\beta }^+(x, t)= {\left\{ \begin{array}{ll} 0 &{} \text {if } t< 0 \\ -\beta t^{p-1} &{} \text {if } 0 \le t < u_0 \\ -\beta u_0^{p-1} &{} \text {if } t \ge u_0 \end{array}\right. }, \end{aligned}$$

(3.6)

and set

$$\begin{aligned}&B^{+}(x, t):= \int _0^t b^{+}(x, s) \;ds \quad \text {as well as} \quad B_{\beta }^{+}(x, t):= \int _0^t b_{\beta }^{+}(x, s) \;ds, \quad \text {and} \nonumber \\&F(x, t):= \int _0^t f(x, s) \;ds. \end{aligned}$$

(3.7)

It is easy to verify that the functions in (3.6) and (3.7) are Carathéodory functions. Moreover we consider the $C^1$-functional $J^{+}:W^{1, {\mathcal {H}}}(\Omega ) \rightarrow {\mathbb {R}}$ defined by

$$\begin{aligned} J^{+}(u)= {\mathscr {M}}\left[ \phi _{{\mathcal {H}}}(\nabla u)\right] +\frac{\vartheta }{p} \Vert u\Vert _p^p+ \frac{1}{q} \Vert u\Vert _{q, a}^q- \int _{\Omega }B^{+}(x, u) \;dx- \int _{\partial \Omega }B_{\beta }^{+}(x, u) \;d\sigma . \end{aligned}$$

(3.8)

We aim to apply the direct methods of the calculus of variations to $J^+$. To this end, we first show that $J^+$ is coercive. Indeed, let $u \in W^{1, {\mathcal {H}}}(\Omega )$ be such that $\Vert u\Vert _{1, {\mathcal {H}}}>1$. Using hypothesis $(M_1)$ and taking into account that $p<q$ we have

$$\begin{aligned} {\mathscr {M}}[\phi _{{\mathcal {H}}}(\nabla u)] \ge \frac{1}{\theta }M[\phi _{{\mathcal {H}}}(\nabla u)] \phi _{{\mathcal {H}}}(\nabla u) \ge \frac{1}{q\theta } M[\phi _{{\mathcal {H}}}(\nabla u)] \varrho _{{\mathcal {H}}}(\nabla u). \end{aligned}$$

Furthermore, we see that, thanks to the truncations (3.6) and hypothesis $(f_1)$, we have that the last two terms in (3.8) are bounded. These facts together with (2.8) yield that $J^+$ is coercive.

We then show that $J^+$ is also (sequentially weakly) lower semicontinuous. Indeed, let us take $u\in W^{1,\mathcal {H}}(\Omega )$ and $(u_j)_j\subset W^{1,\mathcal {H}}(\Omega )$ such that $u_j\rightharpoonup u$ in $W^{1,\mathcal {H}}(\Omega )$. By Propositions 2.1–2.3 and [4, Theorem 4.9] there exists a subsequence, still denoted by $(u_j)_j$, such that, as $j\rightarrow \infty$,

$$\begin{aligned}&u_j\rightarrow u \text{ in } L^{p}(\Omega )\cap L^q_a(\Omega ), \qquad \nabla u_j\rightharpoonup \nabla u \text{ in } \left[ L^{{\mathcal {H}}}(\Omega )\right] ^N,\qquad \phi _{{\mathcal {H}}}(\nabla u_j)\rightarrow \ell ,\\&u_j(x)\rightarrow u(x) \text{ for } \text{ a.e. } x\in \Omega ,\qquad u_j(x) \rightarrow u(x) \text{ for }\ \sigma \text {-a.e. }x\in \partial \Omega . \end{aligned}$$

We use the fact that ${{\mathscr {M}}}$ is increasing and continuous, $\phi _{{\mathcal {H}}}$ is sequentially weakly lower semicontinuous, and $B^+$ and $B^+_\beta$ are Carathéodory functions bounded from below to apply the Fatou’s lemma and achieve

$$\begin{aligned} J^+(u)&\le {{\mathscr {M}}}\left[ \lim _{j\rightarrow \infty }\phi _{{\mathcal {H}}}(\nabla u_j)\right] +\liminf _{j\rightarrow \infty }\left( \frac{\vartheta }{p}\Vert u_j\Vert ^p_p +\frac{1}{q}\Vert u_j\Vert ^q_{q,a}\right) \\&\quad -\int _{\Omega }\lim _{j\rightarrow \infty } B(x,u_j) \; dx- \int _{\partial \Omega }\lim _{j\rightarrow \infty } B^+_\beta (x,u_j) \;d\sigma \\&\le \lim _{j\rightarrow \infty }{{\mathscr {M}}}[\phi _{{\mathcal {H}}}(\nabla u_j)]+\liminf _{j\rightarrow \infty }\left( \frac{\vartheta }{p}\Vert u_j\Vert ^p_p +\frac{1}{q}\Vert u_j\Vert ^q_{q,a}\right) \\&\quad -\liminf _{j\rightarrow \infty } \int _{\Omega }B(x,u_j)\; dx -\liminf _{j\rightarrow \infty }\int _{\partial \Omega }B^+_\beta (x,u_j) \; d\sigma \\&\le \liminf _{j\rightarrow \infty }J^+(u_j). \end{aligned}$$

Therefore, there exists a function $\widetilde{u} \in W^{1, {\mathcal {H}}}(\Omega )$ such that

$$\begin{aligned} J^+(\widetilde{u})= \inf \left\{ J^+(u): \, u \in W^{1, {\mathcal {H}}}(\Omega )\right\} . \end{aligned}$$

(3.9)

Let us verify that $\widetilde{u}$ is not trivial. First of all, thanks to hypothesis $(f_3)$, for all $\varepsilon >0$ there exists $\delta \in (0,1)$ such that

$$\begin{aligned} F(x, t) \le \frac{\varepsilon }{p} |t|^p \quad \text {for a.e. } x \in \Omega \text { and for all } |t| \le \delta . \end{aligned}$$

(3.10)

We further define the function $h:[0,\infty )\rightarrow [M(0),\infty )$ such that

$$\begin{aligned} h(t)={\left\{ \begin{array}{ll}\,M(0)&{}\text { if }t=0\\ \displaystyle \max _{s\in (0,t]} \frac{{{\mathscr {M}}}(s)}{s} &{}\text { otherwise} \end{array}\right. }. \end{aligned}$$

(3.11)

It is easy to see that h is well-defined and continuous. These facts, together with (3.3), guarantee that there exists $t_1\in (0,1)$ such that

$$\begin{aligned} \zeta > \vartheta + \max \{1,h(t_1)\}\lambda _{1, p}. \end{aligned}$$

(3.12)

Let $u_{1, p}$ be the normalized eigenfunction, that is $\Vert u_{1,p}\Vert _p= 1$, corresponding to $\lambda _{1,p}$. Since $u_{1,p} \in \text {int} \left( C^1(\overline{\Omega })_+\right)$, we can choose $t_2 \in (0,t_1]$ sufficiently small so that $t u_{1,p} \in [0, \delta ]$ for all $t\in (0,t_2]$, which implies that

$$\begin{aligned} t u_{1,p} \in (0, u_0)\qquad \text {for all }x \in \overline{\Omega }\text { and }t\in (0,t_2]. \end{aligned}$$

(3.13)

Moreover, let us choose $t_3\in (0,t_2]$ such that

$$\begin{aligned} \phi _{{\mathcal {H}}}(\nabla (tu_{1,p}))\le t_1 \qquad \text {for all}\ t\in [0,t_3]. \end{aligned}$$

(3.14)

Taking (3.11) into account, (3.14) implies that

$$\begin{aligned} {\mathscr {M}}[\phi _{{\mathcal {H}}}(\nabla (t u_{1,p}))] \le h(t_1) \phi _{{\mathcal {H}}}(\nabla (t u_{1,p})) \end{aligned}$$

for every $t\in (0,t_3]$. Then we use the truncations in (3.6) together with (3.10), (3.13), (3.14), and $(f_2)$ to achieve

$$\begin{aligned} J^+({t} u_{1,p})&= {\mathscr {M}} \left[ \phi _{{\mathcal {H}}}(\nabla ({t} u_{1,p}))\right] +\frac{\vartheta }{p} \Vert {t} u_{1,p}\Vert _p^p+ \frac{1}{q} \Vert {t} u_{1,p}\Vert _{q, a}^q \nonumber \\&\qquad - \int _{\Omega }B^+(x, {t} u_{1,p}) \; dx- \int _{\partial \Omega }B_{\beta }^+(x, {t} u_{1,p})\; d\sigma \nonumber \\&\le h(t_1) \frac{{t}^p}{p} \Vert \nabla u_{1,p} \Vert _p^p+ h(t_1) \frac{{t}^q}{q} \Vert \nabla u_{1, p}\Vert _{q, a}^q+ \vartheta \frac{{t}^p}{p}+ \frac{{t}^q}{q} \Vert u_{1,p}\Vert _{q, a}^q \nonumber \\&\qquad -\zeta \frac{{t}^p}{p} + \int _{\Omega }F(x, {t} u_{1,p}) \; dx + \beta \frac{{t}^p}{p} \Vert u_{1,p}\Vert _{p, \partial \Omega }^p \nonumber \\&\le \max \left\{ 1, h(t_1) \right\} \frac{{t}^p}{p} \left( \Vert \nabla u_{1,p}\Vert _p^p+ \beta \Vert u_{1,p}\Vert _{p, \partial \Omega }^p \right) + h(t_1) \frac{{t}^q}{q} \Vert \nabla u_{1,p}\Vert _{q, a}^q \nonumber \\&\qquad + \vartheta \frac{{t}^p}{p}+ \frac{{t}^q}{q} \Vert u_{1,p}\Vert _{q, a}^q- \zeta \frac{{t}^p}{p}+ \varepsilon \frac{{t}^p}{p} \nonumber \\&\le {t}^p \left( \frac{\max \left\{ 1, h(t_1) \right\} \lambda _{1,p} +\vartheta - \zeta + \varepsilon }{p} \right) + {t}^q \left( \frac{h(t_1) \Vert \nabla u_{1,p}\Vert _{q, a}^q+ \Vert u_{1,p}\Vert _{q, a}^q}{q} \right) , \end{aligned}$$

(3.15)

for every $t\in (0,t_3]$. Since $p<q$, we can choose $\overline{t}\in (0,t_3]$ such that

$$\begin{aligned} -\overline{t}^p \left( \frac{\zeta -\vartheta -\max \left\{ 1, h(t_1) \right\} \lambda _{1,p}-\varepsilon }{p} \right) + \overline{t}^q \left( \frac{h(t_1) \Vert \nabla u_{1,p}\Vert _{q, a}^q +\Vert u_{1,p}\Vert _{q, a}^q}{q} \right) <0, \end{aligned}$$

(3.16)

where $\varepsilon :=\frac{1}{2} ({\zeta -\vartheta -\max \left\{ 1, h(t_1) \right\} \lambda _{1,p}})>0$ thanks to (3.12).

From (3.15) it follows that $J^+(\overline{t} u_{1,p})< 0$, therefore, by (3.9) we have

$$\begin{aligned} J^+(\widetilde{u})\le J^+(tu_{1,p})<0= J^+(0), \end{aligned}$$

which implies that $\widetilde{u} \ne 0$. Let us now show the bound $\widetilde{u} \in [0, u_0]$. We observe that (3.9) implies that $\left( J^+\right) '(\widetilde{u})= 0$, that is,

$$\begin{aligned}&M\left[ \phi _{{\mathcal {H}}}(\nabla \widetilde{u})\right] \int _{\Omega }\left( |\nabla \widetilde{u}|^{p-2} \nabla \widetilde{u}+ a(x) |\nabla \widetilde{u}|^{q-2} \nabla \widetilde{u} \right) \cdot \nabla \varphi \; dx \nonumber \\&\qquad + \int _{\Omega }\left( \vartheta |\widetilde{u}|^{p-2} \widetilde{u}+ a(x) |\widetilde{u}|^{q-2} \widetilde{u} \right) \varphi \; dx \nonumber \\&\qquad \quad = \int _{\Omega }b^+(x, \widetilde{u}) \varphi \; dx+ \int _{\partial \Omega }b_{\beta }^+(x, \widetilde{u}) \varphi \; d\sigma \end{aligned}$$

(3.17)

for all $\varphi \in W^{1, {\mathcal {H}}}(\Omega )$. Choosing $\varphi = -\widetilde{u}^- \in W^{1, {\mathcal {H}}}(\Omega )$ in (3.17) and taking into account the truncations (3.6) we get

$$\begin{aligned} M\left[ \phi _{{\mathcal {H}}}(\nabla \widetilde{u})\right] \left( \Vert \nabla \widetilde{u}^-\Vert _p^p+ \Vert \nabla \widetilde{u}^-\Vert _{q, a}^q \right) +\Vert \widetilde{u}^- \Vert _p^p+ \Vert \widetilde{u}^-\Vert _{q, a}^q= 0. \end{aligned}$$

(3.18)

Since all the above terms are nonnegative, the equality in (3.18) is satisfied when $\widetilde{u}^-= 0$, which implies that $\widetilde{u} \ge 0$.

We now choose $\varphi = (\widetilde{u}- u_0)^+ \in W^{1, {\mathcal {H}}}(\Omega )$ in (3.17) and take (3.6) once again into account to achieve

$$\begin{aligned}&M\left[ \phi _{{\mathcal {H}}}(\nabla \widetilde{u})\right] \int _{\Omega }\left( |\nabla \widetilde{u}|^{p-2} \nabla \widetilde{u}+ a(x) |\nabla \widetilde{u}|^{q-2} \nabla \widetilde{u} \right) \cdot \nabla (\widetilde{u}- u_0)^+ \; dx \nonumber \\&\qquad + \int _{\Omega }\left( \vartheta \widetilde{u}^{p-1}+ a(x) \widetilde{u}^{q-1} \right) (\widetilde{u}- u_0)^+ \; dx \nonumber \\&\quad = \int _{\Omega }b^+(x, \widetilde{u}) (\widetilde{u}- u_0)^+ \; dx +\int _{\partial \Omega }b_{\beta }^+(x, \widetilde{u}) (\widetilde{u}- u_0)^+ \; d\sigma \nonumber \\&\quad = \int _{\Omega }(\zeta u_0^{p-1}- f(x, u_0)) (\widetilde{u}- u_0)^+ \; dx+ \int _{\partial \Omega }(-\beta u_0^{p-1}) (\widetilde{u}- u_0)^+ \; d\sigma \nonumber \\&\quad \le 0, \end{aligned}$$

(3.19)

where the last inequality holds by (3.5) and the fact that $u_0>0$. On the one hand, we see that

$$\begin{aligned}&M\left[ \phi _{{\mathcal {H}}}(\nabla \widetilde{u})\right] \int _{\Omega }\left( |\nabla \widetilde{u}|^{p-2} \nabla \widetilde{u}+ a(x) |\nabla \widetilde{u}|^{q-2} \nabla \widetilde{u} \right) \cdot \nabla (\widetilde{u}- u_0)^+ \; dx \nonumber \\&\quad =M\left[ \phi _{{\mathcal {H}}}(\nabla \widetilde{u})\right] \int _{\{\widetilde{u}> u_0\}} |\nabla \widetilde{u}|^p+ a(x) |\nabla \widetilde{u}|^q \; dx, \end{aligned}$$

(3.20)

while on the other hand, exploiting the fact that $\widetilde{u}> u_0> 1$, we get

$$\begin{aligned} 0&\ge \int _{\Omega }(\vartheta \widetilde{u}^{p-1}+ a(x) \widetilde{u}^{q-1}) (\widetilde{u}- u_0)^+ dx \nonumber \\&= \int _{\{\widetilde{u}> u_0\}} \vartheta \widetilde{u}^{p-1} (\widetilde{u}- u_0)+ a(x) \widetilde{u}^{q-1} (\widetilde{u}- u_0) \; dx \nonumber \\&\ge \int _{\{\widetilde{u}> u_0\}} \vartheta (\widetilde{u}- u_0)^p+ a(x) (\widetilde{u}- u_0)^q \; dx. \end{aligned}$$

(3.21)

Gathering (3.19), (3.20) and (3.21), we see that

$$\begin{aligned}{} & {} M\left[ \phi _{{\mathcal {H}}}(\nabla \widetilde{u})\right] \int _{\{\widetilde{u}> u_0\}} |\nabla \widetilde{u}|^p + a(x) |\nabla \widetilde{u}|^q \; dx\\{} & {} \quad + \int _{\{\widetilde{u}> u_0\}} \vartheta (\widetilde{u}- u_0)^p a(x) (\widetilde{u}- u_0)^q \; dx \le 0, \end{aligned}$$

which gives $(\widetilde{u}- u_0)^+= 0$, and therefore $\widetilde{u} \le u_0$. This fact in particular implies that $\widetilde{u} \in L^{\infty }(\Omega )$. By definition of the truncations (3.6), $\widetilde{u}$ turns out to be a weak solution to (3.2).

In order to show the existence of a nonpositive solution, we first fix the constant function $u_1 \equiv -u_0$, then use equation (3.4) and the fact that $p< q$ to achieve

$$\begin{aligned} \zeta |u_1|^{p-2} u_1- f(x, u_1) \ge 0 \quad \text {for a.e. } x \in \Omega . \end{aligned}$$

Then we consider the cut-off, Carathéodory functions $b^-:\Omega \times {\mathbb {R}}\rightarrow {\mathbb {R}}$ and $b^-_{\beta }:\partial \Omega \times {\mathbb {R}}\rightarrow {\mathbb {R}}$ given by

$$\begin{aligned}&b^-(x, t)={\left\{ \begin{array}{ll} \zeta |u_1|^{p-2} u_1- f(x, u_1) &{} \text {if } t \le u_1 \\ \zeta |t|^{p-2} t- f(x, t) &{} \text {if } u_1< t \le 0 \\ 0 &{} \text {if } t> 0 \end{array}\right. },\nonumber \\&b_{\beta }^-(x, t)={\left\{ \begin{array}{ll} -\beta |u_1|^{p-2} u_1 &{} \text {if } t \le u_1 \\ -\beta |t|^{p-2} t &{} \text {if } u_1 < t \le 0 \\ 0 &{} \text {if } t>0 \end{array}\right. }, \end{aligned}$$

(3.22)

set

$$\begin{aligned} B^-(x, t):= \int _0^t b^-(x, s) \; ds \quad \text {as well as} \quad B_{\beta }^-(x, t):= \int _0^t b_{\beta }^-(x, s) \; ds, \end{aligned}$$

(3.23)

and consider the $C^1$-functional $J^-:W^{1, {\mathcal {H}}}(\Omega ) \rightarrow {\mathbb {R}}$ given by

$$\begin{aligned} J^-(u)= {\mathscr {M}}[\phi _{{\mathcal {H}}}(\nabla u)] +\frac{\vartheta }{p} \Vert u\Vert _p^p+ \frac{1}{q} \Vert u\Vert _{q, a}^q- \int _{\Omega }B^-(x, u) \; dx- \int _{\partial \Omega }B_{\beta }^-(x, u) \; d\sigma . \end{aligned}$$

Reasoning in a similar fashion as before we find a global minimizer $\underline{u} \in W^{1, {\mathcal {H}}}(\Omega )$ of $J^-$ such that $\underline{u} \in [u_1, 0]$. By definition of the truncations (3.22) we see that $\underline{u}$ is a nonpositive weak solution to problem (3.2). The proof is complete. $\square$

Now, we consider problem (1.2) with $h_1$, $h_2$ as in (3.1). Then we get

$$\begin{aligned} \left\{ \begin{aligned}&-M\left( \int _{\Omega }|\nabla u|^p dx \right) \Delta _p u- M\left( \int _{\Omega }a(x) |\nabla u|^q dx \right) {{\,\textrm{div}\,}}\left( a(x) |\nabla u|^{q-2} \nabla u\right) \\&\qquad \qquad = (\zeta - \vartheta ) |u|^{p-2} u- a(x) |u|^{q-2} u- f(x, u) \qquad \text {in } \Omega , \\&\left[ M(\left\| \nabla u\right\| ^p_p)|\nabla u|^{p-2}\nabla u+M(\left\| \nabla u\right\| ^q_{q,a})a(x)|\nabla u|^{q-2}\nabla u\right] \cdot \nu = \beta |u|^{p-2} u \qquad \text{ on } \partial \Omega . \end{aligned}\right. \end{aligned}$$

(3.24)

We say that $u \in W^{1, {\mathcal {H}}}(\Omega )$ is a weak solution to (3.24) if

$$\begin{aligned}&M\left( \Vert \nabla u\Vert _p^p\right) \int _{\Omega }|\nabla u|^{p-2} \nabla u \cdot \nabla \varphi \; dx+ M\left( \Vert \nabla u\Vert _{q, a}^q\right) \int _{\Omega }a(x) |\nabla u|^{q-2} \nabla u \cdot \nabla \varphi \; dx \\&\qquad + \int _{\Omega }\left( \vartheta |u|^{p-2} u+ a(x) |u|^{q-2} u \right) \varphi \; dx \\&\quad = \int _{\Omega }\left( \vartheta |u|^{p-2} u- f(x, u)\right) \varphi \; dx- \beta \int _{\partial \Omega }|u|^{p-2} u \varphi \; d\sigma \end{aligned}$$

holds for all $\varphi \in W^{1, {\mathcal {H}}}(\Omega )$. The existence result concerning problem (3.24) reads as follows.

Theorem 3.2

In the same hypotheses of Theorem 3.1, there exist two nontrivial weak solutions $\widetilde{u}$, $\underline{u} \in W^{1, {\mathcal {H}}}(\Omega ) \cap L^{\infty }(\Omega )$ to problem (3.24), such that $\widetilde{u} \ge 0$ and $\underline{u} \le 0$.

Proof

As before, we start by showing the existence of the nonnegative solution, following the proof of Theorem 3.1 until (3.7). Then we consider the $C^1$-functional ${\mathcal {J}}^+:W^{1, {\mathcal {H}}}(\Omega ) \rightarrow {\mathbb {R}}$ given by

$$\begin{aligned} {\mathcal {J}}^+(u)&= \frac{1}{p} {\mathscr {M}} \left( \Vert \nabla u\Vert _p^p\right) + \frac{1}{q} {\mathscr {M}}\left( \Vert \nabla u\Vert _{q, a}^q\right) +\frac{\vartheta }{p} \Vert u\Vert _p^p+ \frac{1}{q} \Vert u\Vert _{q, a}^q\\&\quad -\int _{\Omega }B^+(x, u) \; dx- \int _{\partial \Omega }B_{\beta }^+(x, u) \; d\sigma . \end{aligned}$$

Arguing as in the previous proof and thanks to (2.9) we have that ${\mathcal {J}}^+$ is coercive and lower semicontinuous. Therefore, there exists $\widetilde{u} \in W^{1, {\mathcal {H}}}(\Omega )$ such that

$$\begin{aligned} {\mathcal {J}}^+(\widetilde{u})= \inf \{{\mathcal {J}}^+(u): \, u \in W^{1, {\mathcal {H}}}(\Omega )\}. \end{aligned}$$

In order to show that $\widetilde{u}$ is nontrivial, we let $u_{1,p} \in \text {int} \left( C^1(\overline{\Omega })_+\right)$ be the normalized eigenfunction corresponding to the first eigenvalue $\lambda _{1,p}$ of problem (2.23), and choose $t_1$, $t_2$ as in (3.12) and (3.13), respectively. Moreover, we choose $t_3\in (0,t_2]$ such that

$$\begin{aligned} \Vert \nabla (t u_{1,p}) \Vert _p^p,\, \Vert \nabla (t u_{1,p})\Vert _{q, a}^q\le t_1\qquad \text {for all}\ t\in [0,t_3], \end{aligned}$$

and finally $\overline{t}$ and $\varepsilon >0$ in order to satisfy (3.16). It follows that

$$\begin{aligned} {\mathcal {J}}^+(\widetilde{u}) \le {\mathcal {J}}^+(\overline{t} u_{1,p})< 0={\mathcal {J}}^+(0), \end{aligned}$$

therefore $\widetilde{u} \not \equiv 0$. The proof that $0 \le \widetilde{u} \le u_0$ works as in the previous case, and thus $\widetilde{u}$ is a weak, nonnegative, and bounded solution to (3.24).

In order to find the nonpositive solution we consider the functional ${\mathcal {J}}^-:W^{1, {\mathcal {H}}}(\Omega ) \rightarrow {\mathbb {R}}$ defined by

$$\begin{aligned} {\mathcal {J}}^-(u)= & {} \frac{1}{p} {\mathscr {M}} \left( \Vert \nabla u\Vert _p^p\right) + \frac{1}{q} {\mathscr {M}}\left( \Vert \nabla u\Vert _{q,a}^q\right) +\frac{\vartheta }{p} \Vert u\Vert _p^p+ \frac{1}{q} \Vert u\Vert _{q, a}^q\\{} & {} -\int _{\Omega }B^-(x, u) \; dx- \int _{\partial \Omega }B_{\beta }^-(x, u) \; d\sigma , \end{aligned}$$

with $B^-$ and $B_{\beta }^-$ given as in (3.22), (3.23), and reason as in the previous case. This completes the proof of the theorem. $\square$

We conclude this section pointing out that Theorems 3.1 and 3.2 generalize [10, Theorem 4.2] in a nonlocal Kirchhoff framework. In any case, assumption (3.3) is consistent with the constraint for $\zeta$ requested in [10], when $M\equiv 1$.

4 Infinitely many solutions

In this section we prove the existence of infinitely many solutions to (1.1) and (1.2) when $h_1$ and $h_2$ are symmetric. Throughout this section and differently from the previous one, $\partial \Omega$ is assumed to be only Lipschitz. More precisely, we choose

$$\begin{aligned} h_1(x,t)&=f(x,t)- |t|^{p-2} t- a(x) |t|^{q-2} t \quad \text {for a.e. } x \in \Omega ,\,\text {for all }t\in {\mathbb {R}}, \nonumber \\ h_2(x, t)&= g(x,t) \quad \text {for a.e. } x \in \partial \Omega ,\,\text {for all }t\in {\mathbb {R}}, \end{aligned}$$

(4.1)

where $f:\Omega \times {\mathbb {R}}\rightarrow {\mathbb {R}}$ and $g:\partial \Omega \times {\mathbb {R}}\rightarrow {\mathbb {R}}$ are two Carathéodory functions that satisfy the following assumptions:

$(h_1)$:

there exist exponents $r_1\in (p,p^*)$ and $r_2\in (p,p_*)$, and two constants $c_1$, $c_2>0$ such that

$$\begin{aligned}&|f(x,t)|\le c_1\left( 1+|t|^{r_1-1}\right) \quad \text {for a.e. } x \in \Omega \text { and all }t\in {\mathbb {R}}, \\&|g(x,t)|\le c_2\left( 1+|t|^{r_2-1}\right) \quad \text {for a.e. } x \in \partial \Omega \text { and all }t\in {\mathbb {R}}; \\ \end{aligned}$$

$(h_2)$:

$F(x,t)\ge 0$ and $G(x,t)\ge 0$ for a.e. $x\in \Omega$ and $x\in \partial \Omega$, respectively, and all $t\in {\mathbb {R}}$, where

$$\begin{aligned} F(x,t):=\int _0^tf(x,s)\;ds \quad \text {as well as} \quad G(x,t):=\int _0^tg(x,s)\;ds. \end{aligned}$$

Moreover it holds that

$$\begin{aligned} \lim _{|t|\rightarrow \infty }\frac{F(x,t)}{|t|^{q\theta }}=\infty \quad \text {uniformly for a.e. } x \in \Omega . \\ \end{aligned}$$

$(h_3)$:

It holds that

$$\begin{aligned} {\mathcal {F}}(x,t):= & {} \frac{1}{q\theta }f(x,t) t-F(x,t)\ge 0 \quad \text {as well as} \quad \\ {\mathcal {G}}(x,t):= & {} \frac{1}{q\theta }g(x,t) t-G(x,t)\ge 0 \end{aligned}$$

for a.e. $x\in \Omega$ and a.e. $x \in \partial \Omega$, respectively, and for every $t\in {\mathbb {R}}$ ;

$(h_4)$:

There exist $t_0>0$, $d_1$, $d_2>0$, $s_1>N/p$ and $s_2>(N-1)/(p-1)$ such that

$$\begin{aligned}&F(x,t)^{s_1}\le d_1|t|^{ps_1}{\mathcal {F}}(x,t) \quad \text {for a.e. } x \in \Omega ,\,\text { for all }|t|>t_0, \\&G(x,t)^{s_2}\le d_2|t|^{ps_2}{\mathcal {G}}(x,t) \quad \text {for a.e. } x \in \partial \Omega ,\,\text { for all }|t|>t_0; \\ \end{aligned}$$

$(h_5)$:

$f(x,-t)=-f(x,t)$ and $g(x,-t)=-g(x,t)$ for a.e. $x\in \Omega$ and a.e. $x\in \partial \Omega$, respectively, and all $t\in {\mathbb {R}}$.

Remark 4.1

When $\theta <p_*/q$, two simple model functions f and g satisfying $(h_1)$–$(h_5)$ are given by $f(x,t)=w(x)|t|^{r_1-2}t$ and $g(x,t)=z(x)|t|^{r_2-2}t$, where $w\in L^{\infty }(\Omega )$ with $\inf _\Omega w>0$ and $z\in L^\infty (\partial \Omega )$ with $\inf _{\partial \Omega }z>0$, and with parameters $t_0>0$ and

$$\begin{aligned}&s_1\in \left( \frac{N}{p},\frac{q\theta }{q\theta -p}\right) ,&r_1\in \left( q\theta ,\frac{ps_1}{s_1-1}\right) ,&d_1=\frac{1}{r_1^{s_1}}\cdot \frac{\Vert w\Vert _\infty ^{s_1}}{\inf _\Omega w} \cdot \frac{r_1q\theta }{r_1-q\theta },\nonumber \\&s_2\in \left( \frac{N-1}{p-1},\frac{q\theta }{q\theta -p}\right) ,&r_2\in \left( q\theta ,\frac{ps_2}{s_2-1}\right) ,&d_2=\frac{1}{r_2^{s_2}}\cdot \frac{\Vert z\Vert _{\infty , \partial \Omega }^{s_2}}{\inf _{\partial \Omega } z}\cdot \frac{r_2q\theta }{r_2-q\theta }. \end{aligned}$$

(4.2)

We observe that such parameters in (4.2) exist since $\theta <p_*/q$. Moreover, it can be easily seen that this restriction on the choice of $\theta$ is necessary if we want to have models of this form.

With the choice (4.1) we have that problem (1.1) can be rewritten as

$$\begin{aligned} \left\{ \begin{aligned}&-M\left[ \phi _{{\mathcal {H}}}(\nabla u)\right] \text{ div }\left( |\nabla u|^{p-2}\nabla u+a(x)|\nabla u|^{q-2}\nabla u\right) = f(x, u) - |u|^{p-2} u- a(x) |u|^{q-2} u \quad \text{ in } \Omega ,\\&M\left[ \phi _{{\mathcal {H}}}(\nabla u)\right] \left( |\nabla u|^{p-2} \nabla u+a(x)|\nabla u|^{q-2}\nabla u\right) \cdot \nu = g(x,u) \quad \text{ on } \partial \Omega . \end{aligned}\right. \end{aligned}$$

(4.3)

We say that $u \in W^{1, {\mathcal {H}}}(\Omega )$ is a weak solution to (4.3) if it satisfies

$$\begin{aligned}&M\left[ \phi _{{\mathcal {H}}}(\nabla u)\right] \int _{\Omega }\left( |\nabla u|^{p-2} \nabla u+ a(x) |\nabla u|^{q-2} \nabla u \right) \cdot \nabla \varphi \;dx\\&\qquad + \int _{\Omega }\left( |u|^{p-2}u+ a(x) |u|^{q-2}u \right) \varphi \;dx \\&\quad = \int _{\Omega }f(x, u) \varphi \;dx+ \int _{\partial \Omega }g(x,u) \varphi \;d\sigma \end{aligned}$$

for all $\varphi \in W^{1, {\mathcal {H}}}(\Omega )$. We see that (4.3) is the Euler-Lagrange equation corresponding to the functional $I:W^{1,\mathcal {H}}(\Omega )\rightarrow {\mathbb {R}}$ of class $C^1$ defined as

$$\begin{aligned} I(u):= {\mathscr {M}}\left[ \phi _{{\mathcal {H}}}(\nabla u)\right] +\frac{1}{p} \Vert u\Vert _p^p+ \frac{1}{q} \Vert u\Vert _{q,a}^q- \int _{\Omega }F(x, u) \;dx- \int _{\partial \Omega }G(x, u) \;d\sigma . \end{aligned}$$

Now, we are ready to state the existence result to (4.3).

Theorem 4.2

Let (1.3), $(M_1)$–$(M_2)$, and $(h_1)$–$(h_5)$ hold true. Then, problem (4.3) has infinitely many weak solutions $(u_j)_j$ with unbounded energy.

Theorem 4.2 generalizes [17, Theorem 5.9] where problem (4.3) was considered with $M\equiv 1$, namely without the Kirchhoff coefficient. Here, the function f satisfies the crucial assumption $(h_4)$, different from the quasi-monotonic assumption (1.8) assumed in [17]. Indeed, (1.8) can not work for problem (4.3), even if M satisfies some monotonic condition. This is due to the fact that M in (4.3) depends on $\phi _{{\mathcal {H}}}(\nabla u)$ given in (2.7) and not on the seminorm $\Vert \nabla u\Vert _{{\mathcal {H}}}$. Also, we do not have a proper equivalence between $\phi _{{\mathcal {H}}}(\nabla u)$ and $\Vert \nabla u\Vert _{{\mathcal {H}}}$, but only the following relation

$$\begin{aligned} \frac{1}{q}\min \{\Vert \nabla u\Vert _{{\mathcal {H}}}^p,\Vert \nabla u\Vert _{{\mathcal {H}}}^q\}\le & {} \frac{1}{q}\varrho _{{\mathcal {H}}}(\nabla u)\le \phi _{{\mathcal {H}}}(\nabla u)\\\le & {} \frac{1}{p}\varrho _{{\mathcal {H}}}(\nabla u) \le \frac{1}{p}\max \{\Vert \nabla u\Vert _{{\mathcal {H}}}^p,\Vert \nabla u\Vert _{{\mathcal {H}}}^q\} \end{aligned}$$

implied by Proposition 2.1.

We aim to prove Theorem 4.2 by means of Theorem 2.7. To this end, we first show that I satisfies the (C) condition and then that assumptions (i) and (ii) of Theorem 2.7 are satisfied. We start with the following preliminary result.

Lemma 4.3

Any Cerami sequence $(u_j)_j$ of I is bounded in $W^{1,{\mathcal {H}}}(\Omega )$.

Proof

Let $(u_j)_j$ be a sequence satisfying (2.25) with $E=I$. Hence, there exist $C>0$ and $\varepsilon _j>0$, with $\varepsilon _j\rightarrow 0$, such that

$$\begin{aligned} |\langle I'(u_j),\varphi \rangle |\le \frac{\varepsilon _j \Vert \varphi \Vert _{1,{\mathcal {H}}}}{1+\Vert u_j\Vert _{1,{\mathcal {H}}}} \quad \text{ for } \text{ all } \varphi \in W^{1,{\mathcal {H}}}(\Omega ) \text{ and } j\in {{\mathbb {N}}} \end{aligned}$$

(4.4)

and

$$\begin{aligned} |I(u_j)|\le C\quad \text{ for } \text{ all } j\in {{\mathbb {N}}}. \end{aligned}$$

(4.5)

We claim that $(u_j)_j$ is bounded in $W^{1,{\mathcal {H}}}(\Omega )$.

Arguing by contradiction, we assume that $\Vert u_j\Vert _{1,{\mathcal {H}}}\rightarrow \infty$ as $j\rightarrow \infty$ and, without loss of generality, that $\Vert u_j\Vert _{1,{\mathcal {H}}}> 1$ for j sufficiently large. Set $v_j:=u_j/\Vert u_j\Vert _{1,{\mathcal {H}}}$. It holds that $\Vert v_j\Vert _{1,{\mathcal {H}}}=1$, therefore there exists $v\in W^{1,{\mathcal {H}}}(\Omega )$ such that, up to a subsequence,

$$\begin{aligned} v_j(x)\rightarrow v(x) \text{ for } \text{ a.e. } x\in \Omega ,\qquad v_j\rightarrow v \text{ in } L^{\nu _1}(\Omega )\cap L^{\nu _2}(\partial \Omega ) \end{aligned}$$

(4.6)

for all $\nu _1\in [1,p^*)$ and all $\nu _2\in [1,p_*)$, exploiting the reflexivity of $W^{1,{\mathcal {H}}}(\Omega )$ and Proposition 2.3-(ii) and (iii). We aim to show that $v= 0$. To this end, we set $\Omega ^*=\{x\in \Omega :\,v(x)\ne 0\}$ and assume that $|\Omega ^*|>0$. Since $\Vert u_j\Vert _{1,{\mathcal {H}}}\rightarrow \infty$ as $j\rightarrow \infty$, then

$$\begin{aligned} |u_j|=\big \Vert u_j\big \Vert _{1,{\mathcal {H}}}\cdot |v_j|\rightarrow \infty \quad \text{ a.e. } \text{ in } \Omega ^*. \end{aligned}$$

Taking into account $(h_2)$, we get

$$\begin{aligned} \infty = \lim _{j\rightarrow \infty }\frac{F(x,u_j)}{\Vert u_j\Vert _{1,{\mathcal {H}}}^{q\theta }} =\lim _{j\rightarrow \infty }\frac{F(x,u_j)}{|u_j|^{q\theta }}\cdot |v_j|^{q\theta } \quad \text {for a.e. } x\in \Omega ^*. \end{aligned}$$

Then, Fatou’s lemma gives

$$\begin{aligned} \lim _{j\rightarrow \infty }\int _{\Omega }\frac{F(x,u_j)}{\Vert u_j\Vert _{1,{\mathcal {H}}}^{ q\theta }}\;dx=\lim _{j\rightarrow \infty }\int _{\Omega }\frac{F(x,u_j)|v_j|^{ q\theta }}{|u_j|^{ q\theta }}\;dx=\infty . \end{aligned}$$

(4.7)

On the other hand, we use (2.25) together with the nonnegativity of G first and then (2.16) to achieve

$$\begin{aligned} \int _{\Omega }F(x,u_j)\;dx&\le {\mathscr {M}}[\phi _{{\mathcal {H}}}(\nabla u_j)]+\frac{1}{p}\Vert u_j\Vert _p^p+\frac{1}{q}\Vert u_j\Vert _{q,a}^q+C \\&\le {B_1}(1+\Vert u\Vert _{1, {\mathcal {H}}}^q+\Vert u\Vert _{1, {\mathcal {H}}}^{q\theta })+C \quad \text { for all } j\in {{\mathbb {N}}}. \end{aligned}$$

Dividing by $\Vert u_j\Vert _{1,{\mathcal {H}}}^{q\theta }$, passing to the limit superior as $j \rightarrow \infty$ and recalling that $\theta \ge 1$, we have

$$\begin{aligned} \limsup _{j\rightarrow \infty } \int _{\Omega }\frac{F(x,u_j)}{\Vert u_j\Vert _{1,{\mathcal {H}}}^{q\theta }}\;dx<\infty , \end{aligned}$$

(4.8)

which contradicts (4.7). In conclusion, $\Omega ^*$ has zero measure, that is, $v=0$ a.e. in $\Omega$.

Take now $j \in {\mathbb {N}}$ large enough so that $\Vert u_j\Vert _{1,{\mathcal {H}}}>1$. By hypothesis $(M_1)$, inequality (2.25), together with (2.8) and the fact that $\theta \ge 1$, we have

$$\begin{aligned} C\ge I(u_j)&={\mathscr {M}}[\phi _{{\mathcal {H}}}(\nabla u_j)]+\frac{1}{p}\Vert u_j\Vert _p^p+\frac{1}{q}\Vert u_j\Vert _{q,a}^q\\&\quad -\int _{\Omega }F(x,u_j)\;dx-\int _{\partial \Omega }G(x,u_j)\;d\sigma \\&\ge \frac{1}{\theta }M[\phi _{{\mathcal {H}}}(\nabla u_j)]\phi _{{\mathcal {H}}}(\nabla u_j)+\frac{1}{q}\varrho _{{\mathcal {H}}}(u_j)\\&\quad -\int _{\Omega }F(x,u_j)\;dx-\int _{\partial \Omega }G(x,u_j)\;d\sigma \\&\ge \frac{{A_1}}{q\theta }\Vert u_j\Vert _{1,{\mathcal {H}}}^p-\int _{\Omega } F(x,u_j)\;dx-\int _{\partial \Omega }G(x,u_j) \;d\sigma . \end{aligned}$$

Since $\Vert u_j\Vert _{1,{\mathcal {H}}}\rightarrow \infty$ as $j\rightarrow \infty$, from the previous inequality we have

$$\begin{aligned} \liminf _{j\rightarrow \infty }\left[ \int _{\Omega }\frac{F(x,u_j)}{\Vert u_j\Vert _{1,{\mathcal {H}}}^p}dx +\int _{\partial \Omega }\frac{G(x,u_j)}{\Vert u_j\Vert _{1,{\mathcal {H}}}^p}d\sigma \right] \ge \frac{{A_1}}{q\theta }>0. \end{aligned}$$

(4.9)

We aim to show that

$$\begin{aligned} \lim _{j\rightarrow \infty }\int _\Omega \frac{F(x,u_j)}{\Vert u_j\Vert _{1,{\mathcal {H}}}^p}dx =\lim _{j\rightarrow \infty }\int _{\partial \Omega }\frac{G(x,u_j)}{\Vert u_j \Vert _{1,{\mathcal {H}}}^p}d\sigma =0, \end{aligned}$$

(4.10)

which will eventually contradict (4.9). We first observe that, thanks to hypothesis $(M_1)$ together with (4.4) and (4.5), we have

$$\begin{aligned} C&\ge I(u_j)-\frac{1}{q\theta }\langle I' (u_j),u_j\rangle \nonumber \\&={\mathscr {M}}[\phi _{{\mathcal {H}}}(\nabla u_j)]-\frac{1}{q\theta } M[\phi _{{\mathcal {H}}}(\nabla u_j)]\varrho _{{\mathcal {H}}}(\nabla u_j)\nonumber \\&\quad +\left( \frac{1}{p}-\frac{1}{q\theta }\right) \Vert u_j\Vert _p^p+\left( \frac{1}{q} -\frac{1}{q\theta }\right) \Vert u_j\Vert _{q,a}^q\nonumber \\&\quad -\int _{\Omega }\left[ F(x,u_j)-\frac{1}{q\theta }f(x,u_j)u_j\right] dx -\int _{\partial \Omega }\left[ G(x,u_j)-\frac{1}{q\theta }g(x,u_j)u_j\right] d\sigma \nonumber \\&\ge \int _\Omega {\mathcal {F}}(x,u_j)\;dx+\int _{\partial \Omega }{\mathcal {G}}(x,u_j) \;d\sigma . \end{aligned}$$

(4.11)

For all $a\ge 0$ and $b>a$, we now set $\Omega _j(a,b):=\left\{ x\in \Omega :\,a\le |u_j(x)|<b\right\}$. Thanks to $(h_1)$ and (4.6) it follows that

$$\begin{aligned} \int _{\Omega _j(0,t_0)}\frac{F(x,u_j)}{\Vert u_j\Vert _{1,{\mathcal {H}}}^p}\;dx&\le c_1\int _{\Omega _j(0,t_0)}\left( \frac{|u_j|}{\Vert u_j\Vert _{1,{\mathcal {H}}}^p} +\frac{|u_j|^{r_1}}{{r_1}\Vert u_j\Vert _{1,{\mathcal {H}}}^p}\right) dx\nonumber \\&\le c_1\left( \frac{\Vert u_j\Vert _1}{\Vert u_j\Vert _{1,{\mathcal {H}}}^p} +\frac{1}{r_1}\int _{\Omega _j(0,t_0)}|u_j|^{r_1-p}|v_j|^p\;dx\right) \nonumber \\&\le c_1\left( \frac{C}{\Vert u_j\Vert _{1,{\mathcal {H}}}^{p-1}}+\frac{t_0^{r_1-p}}{r_1} \Vert v_j\Vert _p^p\right) \rightarrow 0\quad \text{ as } j\rightarrow \infty , \end{aligned}$$

(4.12)

being $v=0$. On the other hand, by $(h_4)$, (4.11), and Hölder’s inequality, we have

$$\begin{aligned} \int _{\Omega _j(t_0,\infty )}\frac{F(x,u_j)}{\Vert u_j\Vert _{1,{\mathcal {H}}}^p}\;dx&=\int _{\Omega _j(t_0,\infty )}\frac{F(x,u_j)}{|u_j|^p}|v_j|^p \;dx\nonumber \\&\le \left[ \int _{\Omega _j(t_0,\infty )}\left( \frac{F(x,u_j)}{|u_j|^p}\right) ^{s_1} dx\right] ^{1/s_1} \left( \int _{\Omega _j(t_0,\infty )}|v_j|^{ps_1'} \;dx\right) ^{1/s_1'}\nonumber \\&\le d_1^{1/s_1}\left( \int _{\Omega _j(t_0,\infty )}{\mathcal {F}} (x,u_j) \;dx\right) ^{1/s_1} \Vert v_j\Vert _{ps_1'}^p\nonumber \\&\le d_1^{1/s_1}C^{1/s_1} \Vert v_j\Vert _{ps_1'}^p\rightarrow 0\quad \text{ as } j\rightarrow \infty , \end{aligned}$$

(4.13)

taking once again (4.6) into account, with $v=0$, and since $ps_1'<p^*$ thanks to $(h_4)$. Combining (4.12) and (4.13), we get

$$\begin{aligned} \lim _{j\rightarrow \infty }\int _\Omega \frac{F(x,u_j)}{\Vert u_j\Vert _{1,{\mathcal {H}}}^p}dx=0. \end{aligned}$$

Reasoning in a similar way and exploiting the fact that $ps_2'<p_*$ thanks to hypothesis $(h_4)$, we have

$$\begin{aligned} \lim _{j\rightarrow \infty }\int _{\partial \Omega }\frac{G(x,u_j)}{\Vert u_j \Vert _{1,{\mathcal {H}}}^p}d\sigma =0. \end{aligned}$$

Therefore, (4.10) follows, giving the desired contradiction. This allows us to conclude that $(u_j)_j$ is bounded in $W^{1,{\mathcal {H}}}(\Omega )$. $\square$

Lemma 4.4

The functional I satisfies the (C) condition.

Proof

Let $(u_j)_j\subset W^{1,{\mathcal {H}}}(\Omega )$ be a sequence satisfying (2.25) with $E=I$. Thanks to Lemma 4.3 we have that $(u_j)_j$ is bounded in $W^{1,{\mathcal {H}}}(\Omega )$. Therefore, taking into account Propositions 2.1–2.3 and the reflexivity of $W^{1,{\mathcal {H}}}(\Omega )$, there exist a subsequence, still denoted by $(u_j)_j$, and $u\in W^{1,{\mathcal {H}}}(\Omega )$ such that

$$\begin{aligned}&u_j\rightarrow u \text{ in } L^{{\mathcal {H}}}(\Omega ),\qquad \nabla u_j \rightharpoonup \nabla u \text{ in } \left[ L^{{\mathcal {H}}}(\Omega )\right] ^N, \qquad \phi _{{\mathcal {H}}}(\nabla u_j)\rightarrow \ell ,\nonumber \\&u_j\rightharpoonup u \text{ in } W^{1,\mathcal {H}}(\Omega ),\qquad u_j\rightarrow u \text{ in } L_a^q(\Omega )\cap L^{\nu _1}(\Omega )\cap L^{\nu _2}(\partial \Omega ), \end{aligned}$$

(4.14)

as $j\rightarrow \infty$, with $\nu _1\in [1,p^*)$ and $\nu _2\in [1,p_*)$.

We aim to show that such $(u_j)_j$ is strongly convergent in $W^{1,{\mathcal {H}}}(\Omega )$. Let us distinguish between two possible situations. We first assume that $\ell =0$. Therefore, since $\phi _{{\mathcal {H}}}(v)\ge \varrho _{{\mathcal {H}}}(v)/q\ge 0$ for all $v\in W^{1,{\mathcal {H}}}(\Omega )$, thanks to Proposition 2.1-(v) we have $\nabla u_j\rightarrow \overline{0}$ in $\left[ L^{{\mathcal {H}}}(\Omega )\right] ^N$. Thus we can conclude that $u_j\rightarrow u$ in $W^{1,{\mathcal {H}}}(\Omega )$ as $j\rightarrow \infty$, with u constant a.e. in $\Omega$.

On the other hand, let us suppose $\ell >0$. Thanks to (4.14) and Proposition 2.4 it suffices to show that

$$\begin{aligned} \limsup _{j\rightarrow \infty }\langle L(u_j)-L(u),u_j-u\rangle \le 0, \end{aligned}$$

where L is the functional defined in (2.5). Taking into account hypothesis $(h_1)$, the boundedness of $(u_j)_j$, the convergences in (4.14), and applying Hölder’s inequality we obtain that, as $j\rightarrow \infty$,

$$\begin{aligned} \left| \int _\Omega f(x,u_j)(u_j-u) \;dx\right|&\le c_1\int _\Omega \left( 1+|u_j|^{r_1-1}\right) |u_j-u|\;dx\nonumber \\&\le c_1\left( \Vert u_j-u\Vert _1+\left\| u_j\right\| _{r_1}^{r_1-1} \Vert u_j-u\Vert _{r_1}\right) \rightarrow 0, \end{aligned}$$

(4.15)

as well as

$$\begin{aligned} \left| \int _{\partial \Omega } g(x,u_j)(u_j-u)\;d\sigma \right|&\le c_2\int _{\partial \Omega }\left( 1+|u_j|^{r_2-1}\right) |u_j-u|\;d\sigma \nonumber \\&\le c_2\left( \Vert u_j-u\Vert _{1,\partial \Omega }+\left\| u_j \right\| _{r_2,\partial \Omega }^{r_2-1}\Vert u_j-u\Vert _{r_2,\partial \Omega }\right) \rightarrow 0, \end{aligned}$$

(4.16)

similarly,

$$\begin{aligned} \left| \int _\Omega |u_j|^{p-2}u_j(u_j-u)\;dx\right| \le \Vert u_j\Vert _p^{p-1}\Vert u-u_j\Vert _p\rightarrow 0, \end{aligned}$$

(4.17)

and finally

$$\begin{aligned} \left| \int _\Omega a(x)|u_j|^{q-2}u_j(u_j-u)\;dx\right| \le \Vert u_j\Vert _{q,a}^{q-1}\Vert u-u_j\Vert _{q,a}\rightarrow 0. \end{aligned}$$

(4.18)

Thus, by means of (2.25) and (4.14)–(4.18) we get

$$\begin{aligned} o(1)=\langle I'(u_j),u_j-u\rangle&=M\left[ \phi _{{\mathcal {H}}}(\nabla u_j)\right] \langle L(u_j),u_j-u\rangle \nonumber \\&\quad +\int _\Omega |u_j|^{p-2}u_j(u_j-u)\;dx+\int _\Omega a(x) |u_j|^{q-2}u_j(u_j-u)\;dx\nonumber \\&\quad -\int _\Omega f(x,u_j)(u_j-u)\;dx-\int _{\partial \Omega } g(x,u_j)(u_j-u)\;d\sigma \nonumber \\&=M(\ell )\langle L(u_j),u_j-u\rangle +o(1)\qquad \text { as }j\rightarrow \infty . \end{aligned}$$

(4.19)

Moreover, thanks to Hölder’s inequality and Proposition 2.1, given $\phi \in [L^{\mathcal {H}}(\Omega )]^N$ such that $\Vert \phi \Vert _{\mathcal {H}}=1$ it holds that

$$\begin{aligned}&\left| \int _\Omega \left( |\nabla u|^{p-2}\nabla u+a(x)|\nabla u|^{q-2}\nabla u\right) \cdot \phi \;dx\right| \\&\quad \le \int _{\Omega }|\nabla u|^{p-1}|\phi |\;dx+\int _{\Omega }a (x)^{\frac{q-1}{q}}|\nabla u|^{q-1}a(x)^{\frac{1}{q}}|\phi |\;dx\\&\quad \le \Vert \nabla u\Vert _p^{p-1}\Vert \phi \Vert _p+\Vert \nabla u\Vert _{q,a}^{q-1}\Vert \phi \Vert _{q,a}\\&\quad \le \max \{\Vert \nabla u\Vert _p^{p-1}, \Vert \nabla u\Vert _{q,a}^{q-1}\}\varrho _{\mathcal {H}}(\phi )\\&\quad =\max \{\Vert \nabla u\Vert _p^{p-1},\Vert \nabla u\Vert _{q,a}^{q-1}\}. \end{aligned}$$

This implies that the functional

$$\begin{aligned} P:\phi \in \left[ L^{{\mathcal {H}}}(\Omega )\right] ^N\mapsto \int _\Omega \left( |\nabla u|^{p-2}\nabla u+a(x)|\nabla u|^{q-2}\nabla u\right) \cdot \phi \,dx \end{aligned}$$

is linear and bounded. Therefore, by (4.14) we have

$$\begin{aligned} \langle L(u),u_j-u\rangle =\int _\Omega \left( |\nabla u|^{p-2} \nabla u+a(x)|\nabla u|^{q-2}\nabla u\right) \cdot \nabla (u_j- u) \; dx\rightarrow 0 \text{ as } j\rightarrow \infty . \end{aligned}$$

(4.20)

Combining (4.19)–(4.20), using Proposition 2.4 and taking into account that $M(\ell )>0$ thanks to $(M_2)$, we conclude that $u_j\rightarrow u$ in $W^{1,{\mathcal {H}}}(\Omega )$ as $j\rightarrow \infty$. This completes the proof. $\square$

We now point out that, since $W^{1, {\mathcal {H}}}(\Omega )$ is a reflexive and separable Banach space, there exist two sequences $(e_j)_j \subset W^{1, {\mathcal {H}}}(\Omega )$ and $(e^*_j)_j \subset \left( W^{1, {\mathcal {H}}}(\Omega )\right) ^*$ that satisfy (2.26)–(2.28).

Then we can state the following lemma, which is strongly inspired by [21, Lemma 7.1]. First of all, for all $j \in {\mathbb {N}}$ we set

$$\begin{aligned} \beta _j:=\sup _{u\in Z_j,\,\left\| {u}\right\| _{1,{\mathcal {H}}}=1} \Vert u\Vert _{r_1} \quad \text {as well as} \quad \xi _j:=\sup _{u\in Z_j, \,\left\| {u}\right\| _{1,{\mathcal {H}}}=1} \Vert u\Vert _{r_2,\partial \Omega }, \end{aligned}$$

(4.21)

where $Z_j$ is defined in (2.28) and $r_1$, $r_2$ are chosen as in $(h_1)$.

Lemma 4.5

It holds that

$$\begin{aligned} \lim _{j\rightarrow \infty }\beta _j=\lim _{j\rightarrow \infty }\xi _j=0. \end{aligned}$$

Proof

The claim for $(\beta _j)_j$ is the content of [21, Lemma 7.1], then we are left to show that $\displaystyle \lim _{j\rightarrow \infty }\xi _j=0$. From (4.21), for all $j\in {\mathbb {N}}$ we choose $u_j\in Z_j$ such that $\left\| {u_j}\right\| _{1,{\mathcal {H}}}=1$ and

$$\begin{aligned} \xi _j\le \Vert u_j\Vert _{r_2,\partial \Omega }+\frac{1}{j}. \end{aligned}$$

(4.22)

Since $(u_j)_j$ is bounded in the reflexive space $W^{1,\mathcal {H}}(\Omega )$, thanks to Proposition 2.3–(iii) there exists $u\in W^{1,\mathcal {H}}(\Omega )$ such that

$$\begin{aligned} u_j\rightharpoonup u \text{ in } W^{1,\mathcal {H}}(\Omega ), \qquad u_j\rightarrow u \text{ in } L^{r_2}(\partial \Omega ). \end{aligned}$$

(4.23)

Moreover, fix $k\in {\mathbb {N}}$. From (2.27) it follows that $\langle e^*_k,u_j\rangle =0$, for all $j\in {\mathbb {N}}$ big enough. Then we have

$$\begin{aligned} \langle e^*_k,u\rangle =\lim _{j\rightarrow \infty }\langle e^*_k,u_j\rangle =0 \quad \text{ for } \text{ all } k\in {\mathbb {N}}. \end{aligned}$$

This implies that $u=0$ in $W^{1,\mathcal {H}}(\Omega )$. Then, by (4.23) it holds that $u_j\rightarrow 0$ as $j\rightarrow \infty$ in $L^{r_2}(\partial \Omega )$. Therefore, from (4.22) we have the conclusion. $\square$

We are now ready to prove Theorem 4.2.

Proof of Theorem 4.2

Thanks to hypothesis $(h_5)$ and Lemma 4.4 we have that I is an even functional and satisfies the (C) condition. Thus we are left to verify that conditions (i) and (ii) of Theorem 2.7 hold true. We start with condition (i). By $(h_1)$ we have that

$$\begin{aligned} F(x,t)\le c_1\left( |t|+\frac{|t|^{r_1}}{r_1}\right) \quad \text{ for } \text{ a.e. } x\in \Omega \end{aligned}$$

(4.24)

as well as

$$\begin{aligned} G(x,t)\le c_2\left( |t|+\frac{|t|^{r_2}}{r_2}\right) \quad \text{ for } \text{ a.e. } x\in \partial \Omega , \end{aligned}$$

(4.25)

for all $t \in {\mathbb {R}}$. Thus, for all $u\in Z_j$ with $\left\| {u}\right\| _{1,{\mathcal {H}}}>1$, thanks to (2.8), hypothesis $(M_1)$, (4.21), (4.24), (4.25), and the Hölder’s inequality, we have

$$\begin{aligned} I(u)&\ge \frac{1}{q\theta }\left( M[\phi _{{\mathcal {H}}}(\nabla u)] \phi _{{\mathcal {H}}}(\nabla u)+\varrho _{{\mathcal {H}}}(u)\right) -c_1\left( \Vert u\Vert _{1}+\frac{\Vert u\Vert _{r_1}^{r_1}}{r_1}\right) \nonumber \\&\quad +c_2\left( \left\| u\right\| _{1,\partial \Omega } +\frac{\left\| u\right\| _{r_2,\partial \Omega }^{r_2}}{r_2}\right) \nonumber \\&\ge \frac{{A_1}}{q\theta }\left\| {u}\right\| _{1,{\mathcal {H}}}^p -c_1\left( |\Omega |^{(r_1-1)/r_1}\Vert u\Vert _{r_1} +\frac{\Vert u\Vert _{r_1}^{r_1}}{r_1}\right) \nonumber \\&\quad +c_2\left( \sigma (\Omega )^{(r_2-1)/r_2}\left\| u\right\| _{r_2,\partial \Omega } +\frac{\left\| u\right\| _{r_2,\partial \Omega }^{r_2}}{r_2}\right) \nonumber \\&\ge {C}\left( \left\| {u}\right\| _{1,{\mathcal {H}}}^p-\beta _j\left\| {u}\right\| _{1,{\mathcal {H}}}-\beta _j^{r_1} \left\| {u}\right\| _{1,{\mathcal {H}}}^{r_1}-\xi _j\left\| {u}\right\| _{1,{\mathcal {H}}}-\xi _j^{r_2}\left\| {u}\right\| _{1,{\mathcal {H}}}^{r_2}\right) . \end{aligned}$$

(4.26)

for a suitable $C>0$.

Set now $r:=\max \{r_1,r_2\}>p$ and $\eta _j:=\max \{\beta _j,\xi _j\}$, with $\beta _j$ and $\xi _j$ given in (4.21). By Lemma 4.5, we have that $\eta _j<1$ if $j\in {\mathbb {N}}$ is sufficiently large. Hence, by (4.26), for all $u\in Z_j$ with $\left\| {u}\right\| _{1,{\mathcal {H}}}>1$ and $j\in {\mathbb {N}}$ big enough, we get

$$\begin{aligned} I(u)\ge {C}(1-4\eta _j\left\| {u}\right\| _{1,{\mathcal {H}}}^{r-p})\left\| {u}\right\| _{1,{\mathcal {H}}}^p. \end{aligned}$$

(4.27)

Let us choose

$$\begin{aligned} \gamma _j:=\left( \frac{1}{8\eta _j}\right) ^\frac{1}{r-p}. \end{aligned}$$

Then $\gamma _j\rightarrow \infty$ as $j\rightarrow \infty$, since $\eta _j\rightarrow 0$ as $j\rightarrow \infty$ thanks to Lemma 4.5 and the fact that $r>p$. Inequality (4.27) yields that, for all $u\in Z_j$ with $\left\| {u}\right\| _{1,{\mathcal {H}}}=\gamma _j$,

$$\begin{aligned} I(u)\ge \frac{\widetilde{C}}{2}\gamma _j^p\rightarrow \infty \quad \text{ as } j\rightarrow \infty , \end{aligned}$$

which gives the validity of condition (i).

In order to prove condition (ii) we argue by contradiction. Therefore we assume that there exists $j>0$ such that for each $n\in {{\mathbb {N}}}{\setminus }\{0\}$ we can find a function $u_n\in Y_j$ with $\Vert u_n\Vert _{1,{{\mathcal {H}}}}>n$ and $I(u_n)>0$. Let us define, for each n, $v_n:=u_n/\Vert u_n\Vert _{1,{{\mathcal {H}}}}$. Of course, $\Vert v_n\Vert _{1,{{\mathcal {H}}}}=1$. Since $(Y_j,\Vert \cdot \Vert _{1,{{\mathcal {H}}}})$ is a finite dimensional Banach space, it follows that there exists $v\in Y_j$ such that, up to a subsequence,

$$\begin{aligned} \Vert v_n-v\Vert _{1,{{\mathcal {H}}}}\rightarrow 0 \end{aligned}$$

as $n \rightarrow \infty$. Since also $\Vert v\Vert _{1,{{\mathcal {H}}}}=1$, setting $A:=\{x\in \Omega :\,v(x)\ne 0\}$, it follows that $|A|\ne 0$. Therefore, arguing as in the proof of Lemma 4.3, thanks to $(h_2)$ we conclude that

$$\begin{aligned} \lim _{n\rightarrow \infty }\int _A\frac{F(x,u_n(x))}{\Vert u_n \Vert _{1,{{\mathcal {H}}}}^{q\theta }}\;dx=+\infty . \end{aligned}$$

(4.28)

On the other hand, as $I(u_n)>0$, and thanks to the nonnegativity of G and (2.16), we have that

$$\begin{aligned} \int _AF(x,u_n(x))\;dx&\le \int _\Omega F(x,u_n(x))\;dx \le \int _\Omega F(x,u_n(x))\;dx+I(u_n)\nonumber \\&={\mathscr {M}}\left[ \phi _{{\mathcal {H}}}(\nabla u_n)\right] +\frac{1}{p} \Vert u_n\Vert _p^p+ \frac{1}{q} \Vert u_n\Vert _{q,a}^q -\int _{\partial \Omega }G(x, u_n) \;d\sigma \nonumber \\&\le {\mathscr {M}}\left[ \phi _{{\mathcal {H}}}(\nabla u_n)\right] +\frac{1}{p} \Vert u_n\Vert _p^p+ \frac{1}{q} \Vert u_n\Vert _{q,a}^q\nonumber \\&\le B_1(1+\Vert u_n\Vert _{1, {\mathcal {H}}}^q+\Vert u_n\Vert _{1, {\mathcal {H}}}^{q\theta }). \end{aligned}$$

(4.29)

Dividing both extremes of the previous inequality by $\Vert u_n\Vert ^{q\theta }_{1,{{\mathcal {H}}}}$ and passing to the limit, a contradiction with (4.28) follows.

Thus, we can apply Theorem 2.7 to obtain a sequence of critical points of I with unbounded energy. The proof is thus complete. $\square$

We now aim to show existence of infinitely many solutions to (1.2), where $h_1$ and $h_2$ are chosen as in (4.1), that is,

$$\begin{aligned} \left\{ \begin{aligned}&-M\left( \Vert \nabla u\Vert _p^p\right) \Delta _p u-M\left( \Vert \nabla u\Vert _{q,a}^q\right) \text{ div }\left( a(x)|\nabla u|^{q-2}\nabla u\right) \\&\quad = f(x, u) - |u|^{p-2} u- a(x) |u|^{q-2} u \quad \text{ in } \Omega ,\\&\left[ M\left( \Vert \nabla u\Vert _p^p\right) |\nabla u|^{p-2}\nabla u+M \left( \Vert \nabla u\Vert _{q,a}^q\right) a(x)|\nabla u|^{q-2}\nabla u\right] \cdot \nu = g(x,u) \quad \text{ on } \partial \Omega . \end{aligned}\right. \end{aligned}$$

(4.30)

We say that a function $u\in W^{1,{\mathcal {H}}}(\Omega )$ is a weak solution to (4.30) if

$$\begin{aligned}&M(\Vert \nabla u\Vert _p^p)\int _\Omega |\nabla u|^{p-2}\nabla u\cdot \nabla \varphi \;dx +M(\Vert \nabla u\Vert _{q,a}^q)\\&\qquad \int _\Omega a(x)|\nabla u|^{q-2}\nabla u\cdot \nabla \varphi \;dx + \int _{\Omega }\left( |u|^{p-2}u+ a(x) |u|^{q-2}u \right) \varphi \;dx\\&\quad = \int _{\Omega }f(x, u) \varphi \;dx+ \int _{\partial \Omega }g(x,u) \varphi \;d\sigma , \end{aligned}$$

is satisfied for all $\varphi \in W^{1,{\mathcal {H}}}(\Omega )$. In this case, (4.30) is the Euler-Lagrange equation associated with the energy functional ${\mathcal {I}}:W^{1,{\mathcal {H}}}(\Omega )\rightarrow {\mathbb {R}}$ given by

$$\begin{aligned} {\mathcal {I}}(u)&:=\frac{1}{p}{\mathscr {M}}(\Vert \nabla u\Vert _p^p) +\frac{1}{q}{\mathscr {M}}(\Vert \nabla u\Vert _{q,a}^q) +\frac{1}{p}\Vert u\Vert _p^p+\frac{1}{q}\Vert u\Vert _{q,a}^q\\&\quad -\int _\Omega F(x,u)\;dx-\int _{\partial \Omega } G(x,u)\;d\sigma . \end{aligned}$$

Our multiplicity result for (4.30) reads as follows.

Theorem 4.6

Let (1.3), $(M_1)$–$(M_2)$, and $(h_1)$–$(h_5)$ hold true. Then, problem (4.30) has infinitely many weak solutions $(u_j)_j$ with unbounded energy.

Here, we point out that assumption $(h_3)$ could allow us to prove the boundedness of a Palais-Smale sequence for ${\mathcal {I}}$, but just in $W^{1,p}(\Omega )$. This fact is not enough considering ${\mathcal {I}}$ set in $W^{1,{\mathcal {H}}}(\Omega )$, with $W^{1,{\mathcal {H}}}(\Omega )\hookrightarrow W^{1,p}(\Omega )$ by Proposition 2.3-(i). For this reason, we exploit the same ideas employed in the proof of Theorem 4.2 in order to prove Theorem 4.6. That is, we start by showing that Cerami sequences of ${\mathcal {I}}$ are bounded. Then we use this property to show that ${\mathcal {I}}$ satisfies the (C) condition. Finally we apply Theorem 2.7 to ${\mathcal {I}}$.

Lemma 4.7

Any Cerami sequence of ${\mathcal {I}}$ is bounded in $W^{1,{\mathcal {H}}}(\Omega )$.

Proof

The proof works exactly as in Lemma 4.4, with (4.8) and (4.11) following from (2.9) and (2.17), respectively. $\square$

Lemma 4.8

The functional ${\mathcal {I}}$ satisfies the (C) condition.

Proof

Let $(u_j)_j\subset W^{1,{\mathcal {H}}}(\Omega )$ be a sequence satisfying (2.25) with $E={\mathcal {I}}$. Thanks to Lemma 4.7, $(u_j)_j$ is bounded in $W^{1,{\mathcal {H}}}(\Omega )$. Hence, by Propositions 2.1–2.3 and the reflexivity of $W^{1,{\mathcal {H}}}(\Omega )$, there exist a subsequence, still denoted by $(u_j)_j$, and $u\in W^{1,{\mathcal {H}}}(\Omega )$ such that

$$\begin{aligned}&u_j\rightarrow u \text{ in } L^{{\mathcal {H}}}(\Omega ), \qquad \nabla u_j\rightharpoonup \nabla u \text{ in } \left[ L^{{\mathcal {H}}}(\Omega )\right] ^N, \ \Vert \nabla u_j\Vert _p\rightarrow \ell _p,\nonumber \\&\Vert \nabla u_j\Vert _{q,a}\rightarrow \ell _q,\qquad u_j \rightharpoonup u \text{ in } W^{1,\mathcal {H}}(\Omega ),\qquad u_j \rightarrow u \text{ in } L^{\nu _1}(\Omega )\cap L^{\nu _2}(\partial \Omega ), \end{aligned}$$

(4.31)

as $j\rightarrow \infty$, with $\nu _1\in [1,p^*)$ and $\nu _2\in [1,p_*)$. We apply (2.25) together with (4.15)–(4.18) and (4.31), to have

$$\begin{aligned} o(1)=\langle {\mathcal {I}}'(u_j),u_j-u\rangle&=M(\Vert \nabla u_j\Vert _p^p)\int _\Omega |\nabla u_j|^{p-2} \nabla u_j\cdot (\nabla u_j-\nabla u)\;dx\nonumber \\&\quad +M(\Vert \nabla u_j\Vert _{q,a}^q)\int _\Omega a(x)| \nabla u_j|^{q-2}\nabla u_j\cdot (\nabla u_j-\nabla u)\;dx\nonumber \\&\quad +\int _\Omega |u_j|^{p-2}u_j(u_j-u)\;dx+\int _\Omega a(x) |u_j|^{q-2}u_j(u_j-u)\;dx\nonumber \\&\quad -\int _\Omega f(x,u_j)(u_j-u)\;dx-\int _{\partial \Omega } g(x,u_j)(u_j-u)\;d\sigma \nonumber \\&=M(\ell _p^p)\int _\Omega |\nabla u_j|^{p-2}\nabla u_j \cdot (\nabla u_j-\nabla u)\;dx\nonumber \\&\quad +M(\ell _q^q)\int _\Omega a(x)|\nabla u_j|^{q-2} \nabla u_j \cdot (\nabla u_j{-}\nabla u)\;dx{+}o(1)\quad \text {as}\ j{\rightarrow }\infty . \end{aligned}$$

(4.32)

We need now to distinguish between two situations, that depend on the behavior of M at zero.

Case 1: Let M verify $M(0)=0$.

Since $\ell _p\ge 0$ and $\ell _q\ge 0$ in (4.31), we further distinguish among four subcases.

Subcase 1.1: Let $\ell _p=0$ and $\ell _q=0$.

By (4.31) we have $\Vert \nabla u_j\Vert _p\rightarrow 0$ and $\Vert \nabla u_j\Vert _{q,a}\rightarrow 0$ as $j\rightarrow \infty$, which thanks to (2.1) and Proposition 2.1 implies that $\nabla u_j\rightarrow \overline{0}$ in $[L^{{\mathcal {H}}}(\Omega )]^N$. Hence, $u_j\rightarrow u$ in $W^{1,{\mathcal {H}}}(\Omega )$ as $j\rightarrow \infty$, with u constant a.e. in $\Omega$.

Subcase 1.2: Let $\ell _p=0$ and $\ell _q>0$.

From (4.32) and $(M_2)$ we have

$$\begin{aligned} \lim _{j\rightarrow \infty }\int _\Omega a(x)|\nabla u_j|^{q-2}\nabla u_j \cdot (\nabla u_j-\nabla u)\;dx=0. \end{aligned}$$

(4.33)

Moreover, since $(\nabla u_j)_j$ is bounded in $(L^p(\Omega ))^N$ thanks to (4.31), it follows that

$$\begin{aligned} \limsup _{j\rightarrow \infty }\left| \int _\Omega |\nabla u_j|^{p-2}\nabla u_j \cdot (\nabla u_j-\nabla u)\;dx\right|&\le \limsup _{j\rightarrow \infty }\Vert \nabla u_j\Vert _p^{p-1}\Vert \nabla u_j-\nabla u\Vert _p\\&\le \ell _p^{p-1} \limsup _{j\rightarrow \infty }\Vert \nabla u_j-\nabla u\Vert _p=0. \end{aligned}$$

Then we have

$$\begin{aligned} \lim _{j\rightarrow \infty }\langle L(u_j),u_j-u\rangle =0. \end{aligned}$$

This fact, together with (4.20) and Proposition 2.4, allows to conclude that $u_j\rightarrow u$ in $W^{1,\mathcal {H}}(\Omega )$.

Subcase 1.3: Let $\ell _p>0$ and $\ell _q=0$.

The proof works exactly as in Subcase 1.2, hence we omit the details.

Subcase 1.4: Let $\ell _p>0$ and $\ell _q>0$.

It follows that

$$\begin{aligned} \lim _{j\rightarrow \infty }\int _\Omega |\nabla u|^{p-2}\nabla u\cdot \nabla (u_j- u) \;dx=0, \end{aligned}$$

(4.34)

as well as

$$\begin{aligned} \lim _{j\rightarrow \infty }\int _{\Omega }a(x)|\nabla u|^{q-2}\nabla u\cdot \nabla (u_j- u)\;dx=0. \end{aligned}$$

(4.35)

Therefore (4.32), (4.34), and (4.35) yield

$$\begin{aligned}&M(\ell _p^p)\int _\Omega \left( |\nabla u_j|^{p-2}\nabla u_j -|\nabla u|^{p-2}\nabla u\right) \cdot (\nabla u_j-\nabla u)\;dx\nonumber \\&\quad +M(\ell _q^q)\int _\Omega a(x)\left( |\nabla u_j|^{q-2}\nabla u_j -|\nabla u|^{q-2}\nabla u\right) \cdot (\nabla u_j-\nabla u)\;dx=o(1)\nonumber \\&\quad \text {as}\ j\rightarrow \infty . \end{aligned}$$

(4.36)

Exploiting the convexity of the two Laplacian operators of p and q types and the fact that $a(x)\ge 0$ a.e. in $\Omega$ thanks to (1.3), we get

$$\begin{aligned}&\left( |\nabla u_j|^{p-2}\nabla u_j-|\nabla u|^{p-2}\nabla u\right) \cdot (\nabla u_j-\nabla u) \ge 0 \text{ a.e. } \text{ in } \Omega ,\\&a(x)\left( |\nabla u_j|^{q-2}\nabla u_j-|\nabla u|^{q-2}\nabla u\right) \cdot (\nabla u_j-\nabla u) \ge 0 \text{ a.e. } \text{ in } \Omega . \end{aligned}$$

Therefore, from (4.36) we have

$$\begin{aligned} \min \left\{ M(\ell _p^p),M(\ell _q^q)\right\} \limsup _{j\rightarrow \infty }\langle L(u_j)-L(u),u_j-u\rangle \le 0, \end{aligned}$$

being $M(\ell _p^p)$, $M(\ell _q^q)>0$ thanks to $(M_2)$. Hence we can use Proposition 2.4 and (4.31) to conclude that $u_j\rightarrow u$ in $W^{1,{\mathcal {H}}}(\Omega )$ as $j\rightarrow \infty$. This completes the proof of Case 1.

Case 2: Let M verify $M(0)>0$.

Since by $(M_2)$ we have that $M(\ell _p^p)$, $M(\ell _q^q)>0$ for $\ell _p$, $\ell _q\ge 0$, we can argue exactly as in Subcase 1.4 to get the conclusion. $\square$

Proof of Theorem 4.6

The proof works exactly as for Theorem 4.2, with (4.27) and (4.29) following from (2.9) and (2.17), respectively.$\square$