The eigenvalue problem for Kirchhoff-type operators in Musielak–Orlicz spaces

Abstract

Given a Musielak–Orlicz function $\varphi (x,s):\Omega \times [0,\infty )\rightarrow {\mathbb R}$ on a bounded regular domain $\Omega \subset {\mathbb R}^n$ and a continuous function $M:[0,\infty )\rightarrow (0,\infty )$, we show that the eigenvalue problem for the elliptic Kirchhoff’s equation $-M\left( \int \limits _{\Omega }\varphi (x,|\nabla u(x)|)\textrm{d}x\right) \text {div}\left( \frac{\partial \varphi }{\partial s}(x,|\nabla u(x)|)\frac{\nabla u(x)}{|\nabla u(x)|}\right) =\lambda \frac{\partial \varphi }{\partial s}(x,|u(x)|)\frac{u(x)}{|u(x)|}$ has infinitely many solutions in the Sobolev space $W_0^{1,\varphi }(\Omega )$. No conditions on $\varphi$ are required beyond those that guarantee the compactness of the Sobolev embedding theorem.

1 Introduction

In 1883, G. Kirchhoff [13] noted that the vibration of an elastic, variable-length string is modeled by means of the following variant of the classical wave equation:

$$\begin{aligned} \frac{\partial ^2 u}{\partial t^2}=M\left( \int _0^1\left( \frac{\partial u}{\partial x}\right) ^2\right) \frac{\partial ^2 u}{\partial x^2}, \end{aligned}$$

(1.1)

where $M:[0,\infty )\longrightarrow [0,\infty )$ is a suitable increasing function. Since then, a vast amount of literature was devoted to studying the solvability of various Kirchhoff-type equations [2].

In higher dimensions, (1.1) takes up the form

$$\begin{aligned} \frac{\partial ^2 u}{\partial t^2}=M\left( \int \limits _{\Omega }|\nabla u(y)|^2dy\right) \Delta u. \end{aligned}$$

(1.2)

The stationary problem

$$\begin{aligned} {\left\{ \begin{array}{ll} M\left( \int \limits _{\Omega }|\nabla u(y)|^2dy\right) \Delta u=f(x,u)\\ u|_{\partial \Omega }=0 \end{array}\right. } \end{aligned}$$

(1.3)

has been extensively studied under different assumptions on M and f, see for example [1, 19, 20, 26, 27] and the references therein.

Of particular interest is the extension of (1.2) to equations involving the p-Laplacian [10, 18]: If $1<p<\infty$ is a real number and $M\ge 0$ is a continuous function, the p-Kirchhoff operator is defined as

$$\begin{aligned} K_p:W^{1,p}_0(\Omega )\longrightarrow \left( W^{1,p}_0(\Omega )\right) ^* K_p(u)=-M\left( \int \limits _{\Omega }|\nabla u(y)|^pdy\right) \text {div}\left( {|\nabla u(x)|^{p-2}}{\nabla u(x)}\right) , \end{aligned}$$

(1.4)

which clearly generalizes the right-hand side of (1.1). In (1.4), $W^{1,p}_0(\Omega )$ stands for the closure of $C_0^{\infty }(\Omega )$ in the usual Sobolev space $W^{1,p(\cdot )}(\Omega )$. The reader is referred to Sect. 3 for the precise terminology to be used in this work.

From the physical point of view, this operator arose from the need of finding a mathematical model for the motion of a vibrating string under less stringent assumptions than those assumed for the classical derivation of the linear wave equation. Specifically, the linear wave equation is obtained under the assumption that the length of the vibrating string remains constant during the motion. By removing this assumption, the nonlinear operator $K_p$ comes into the play.

Various boundary value problems associated to the p-Kirchhoff operator (1.4) have been studied for example in [10, 18, 19]. The emergence of the variable exponent Lebesgue spaces and the subsequent realization of their role in applications [5] sparked interest in the study of boundary value problems of the type

$$\begin{aligned} {\left\{ \begin{array}{ll} K_p(u)=f(x,u)\\ u|_{\partial \Omega }=0, \end{array}\right. } \end{aligned}$$

(1.5)

for a variable exponent $p=p(x)$.

The variability of the exponent opens a new class of highly non-trivial difficulties, mainly related to its modular nature, that is to its direct relation to the functional

$$\begin{aligned}u\longrightarrow \int _{\Omega } |u(x)|^{p(x)}\textrm{d}x \end{aligned}$$

rather than with the norm

$$\begin{aligned}u\longrightarrow \Vert u\Vert _{p(x)}, \end{aligned}$$

as discussed in [2, 21]. A vast amount of literature exists on boundary value problems of the type (1.5), under the assumption of the variability of the exponent p(x). We refer the reader to some of the most significant from the point of view of the present work, such as [2,3,4, 6,7,8, 21, 23].

In this article we observe that the treatment of a wide class of eigenvalue problem for Kirchhoff-type operators, including, but not limited to the variable-exponent case, can be unified by the consideration of Musielak–Orlicz spaces. With this objective in mind, we study the eigenvalue problem for a general Kirchhoff equation in this framework. In fact, given a suitable Musielak–Orlicz (MO)-function $\varphi$ and an appropriate function M (we refer the reader to the next section for a detailed account of the notation and terminology), the generalized Kirchhoff operator is naturally given by

$$\begin{aligned} K(u)=-M\left( \rho _{\varphi }(|\nabla u|)\right) \text {div}\left( \frac{\partial \varphi }{\partial s}(x,|\nabla u(x)|)\frac{\nabla u(x)}{|\nabla u(x)|}\right) . \end{aligned}$$

(1.6)

We provide a characterization of the first eigenvalue for the operator (1.6) via a Musielak–Orlicz Sobolev embedding theorem that has been obtained in [16, Theorem 5.1].

The present work is organized as follows. In the next section we introduce the notation and terminology to be used in the exposition and present a brief survey on the literature. In Section 3 the definition and basic properties of the Musielak–Orlicz spaces needed in the sequel are given. Section 4 is a brief survey on the Sobolev embedding theorems in the context of Musielak–Orlicz spaces. In Section 5 we delve into some natural properties of the Musielak–Orlicz operators to be considered later and the functional analytic stage is set for the treatment of the eigenvalue problems developed in detail in Section 7. Section 8 contains applications to the variable exponent case, i.e., to the case $\varphi (x,t)=t^{p(x)}$.

2 Known results

In the sequel, $\Omega \subset {\mathbb R}^n$ will denote a bounded domain with a regular boundary (the cone condition will do) and ${{\mathcal {M}}}(\Omega )$ will stand for the vector space of all real-valued, Borel-measurable functions defined on $\Omega$. The subset of ${{\mathcal {M}}}$ consisting of functions

$$\begin{aligned} p:\Omega \longrightarrow [1,\infty ) \end{aligned}$$

will be denoted by ${{\mathcal {P}}}(\Omega )$. The Lebesgue measure of a subset $A\subset {\mathbb R}^n$ will be denoted by |A|.

For $p\in {{\mathcal {P}}}(\Omega )$, the following notation will be used throughout this work:

$$\begin{aligned} p_-:=\text { essinf }_{\Omega }\,p,\,\,\,p_+:=\text { esssup }_{\Omega }\,p. \end{aligned}$$

For $p\in {{\mathcal {P}}}(\Omega )$, the eigenvalue problem

$$\begin{aligned} {\left\{ \begin{array}{ll} K_p(u)=\lambda |u|^{q-2}u\,\,\,\text {in}\,\,\Omega \\ u|_{\partial \Omega }=0 \end{array}\right. } \end{aligned}$$

(2.1)

was studied in [2] for M subject to

$$\begin{aligned} m_1t^{\alpha -1}\le M(t)\le m_2t^{\beta -1}, \end{aligned}$$

(2.2)

for $\beta \ge \alpha > 1$, $m_2\ge m_1> 0$ and variable exponents $p, q\in C({\overline{\Omega }})$ satisfying either [2, Theorem 3.1, Theorem 3.4, Theorem 3.6]

$$\begin{aligned} \beta p_+< q_-\le q_+<p^*, \end{aligned}$$

(2.3)

$$\begin{aligned} 1< q_-\le q_+<\alpha p_-, \end{aligned}$$

(2.4)

or

$$\begin{aligned} 1<q(x)<p(x)<p^*(x) \end{aligned}$$

(2.5)

in $\Omega$. Here

$$\begin{aligned} p^{*}(x)=\frac{np(x)}{n-p(x)}{} \textbf{1}_{(1, n)}(p(x))+\infty \textbf{1}_{[n,\infty )}(p(x)) \end{aligned}$$

and $\textbf{1}_A$ stands for the characteristic function of the set A.

An anisotropic variant of (2.1) was considered in [23], whereas [4] deals with the following weighted version of (2.1):

$$\begin{aligned} {\left\{ \begin{array}{ll} K_p(u)=\lambda V |u|^{q-2}u \\ u|_{\partial \Omega }=0, \end{array}\right. } \end{aligned}$$

(2.6)

for $0\le V\in L^{\infty }(\Omega )$, $p_+<n$ and M subject to (2.2) [4, Theorem 1.4]. The generalized version of (1.5) given by

$$\begin{aligned} {\left\{ \begin{array}{ll} K_p(u)=B\left( \int \limits _{\Omega }\int _0^{|u|}f(x,s)ds\right) f(x,u)\\ u|_{\partial \Omega }=0 \end{array}\right. } \end{aligned}$$

(2.7)

is studied in [12]. Specifically, problem (2.7) is shown to have a solution in $W_0^{1,p(\cdot )}(\Omega )$ under the following assumptions [12, Theorem 3.1]:

• (i) $\int _0^tM(s)ds\ge mt^{\alpha _1}$, $m>0$, for sufficiently large t,

• (ii) for some positive constants $c_1, c_2$ the Carathéodory function

  $$\begin{aligned} f:\Omega \times {\mathbb R}\rightarrow {\mathbb R} \end{aligned}$$

  satisfies the bound $|f(x,t)|\le c_1+c_2|t|^{q(x)}$ for $q\in C({\overline{\Omega }})$, $1<q(x)<p^*(x)$,

• (iii) for some positive constants $A_1, A_2$,

  $$\begin{aligned} \int ^t_{0}B(s)ds\le A_1+A_2t^{\beta _1}, \end{aligned}$$
• (iv)
  $$\begin{aligned} \beta _1q_+<\alpha _1p_-. \end{aligned}$$

For $M(t)=a+b\gamma t^{\gamma -1}$, the study of the solvability of a hyperbolic equation related to the operator $K_p$ can be found in [3]. A polyharmonic version of (1.5) was studied in [6].

In [8], a discussion of problem 1.5 is presented for a linear function $M(t)=a+bt$ [8, Theorem 1.1], whereas in [7] the existence of a solution of (1.5) is proved provided, among other conditions, that $M(t)\ge m_0>0$ for $t>0$ [7, Theorems 3.1$-$3.4].

Associated to every Musielak–Orlicz function $\varphi$, the so-called Matuszewska index of $\varphi$ (see [17]) generalizes the role of the exponent p in the classical Lebesgue spaces; in particular the exponent p is easily verified to be the Matuszewska index of the MO function given by

$$\begin{aligned} (x,t)\longrightarrow t^{p(x)}. \end{aligned}$$

As is observed in [16], sharp conditions (trivially satisfied by the exponent p for the Sobolev embedding stated in [11, 15]) on the Matuszewska index of the MO function $\varphi$ guarantee the compactness of the Sobolev embedding

$$\begin{aligned} W^{1,\varphi }_0(\Omega )\hookrightarrow L^{\varphi }(\Omega ) \end{aligned}$$

(2.8)

for a bounded domain $\Omega \subset {\mathbb R}^n$. Via the compactness of the Sobolev embedding, a natural characterization of the first eigenvalue of the Kirchhoff’s operator can be given, and the results outlined above can be regarded as particular cases of our more general approach, which allows for less stringent conditions than the ones stated in the first part of this Section.

3 Musielak–Orlicz spaces

Throughout this paper $\Omega \subset {\mathbb R}^n$, $n\ge 1$ will stand for a bounded, Lipschitz domain. A convex, left-continuous function

$$\begin{aligned} \varphi :[0,\infty )\longrightarrow [0,\infty ) \end{aligned}$$

with $\varphi (0)=0$, $\lim _{x\rightarrow \infty }\varphi (x)=\infty$ and $\lim _{x\rightarrow 0^+}\varphi (x)=0$ will be said to be an Orlicz function. In particular, any Orlicz function is non-decreasing. The term generalized Orlicz function or Musielak–Orlicz (MO) function will refer to a function

$$\begin{aligned} \varphi :\Omega \times [0,\infty )\rightarrow [0,\infty ) \end{aligned}$$

such that

$$\begin{aligned} \varphi (x,\cdot ):[0, \infty )\rightarrow [0,\infty ) \end{aligned}$$

is an Orlicz function for each fixed $x\in \Omega$ and

$$\begin{aligned} \varphi (\cdot ,y):\Omega \rightarrow [0,\infty ) \end{aligned}$$

is Lebesgue measurable for each fixed $y \in {\mathbb R}$.

The Musielak–Orlicz space $L^{\varphi }(\Omega )$, [24, 25], is the real-vector space $X_{\varphi }$ of all extended-real valued, Borel-measurable functions u on $\Omega$ for which

$$\begin{aligned} \int \limits _{\Omega }\varphi (x,\lambda |u(x)|)\,\textrm{d}x<\infty \,\,\,\text {for some}\,\,\lambda >0, \end{aligned}$$

furnished with the norm

$$\begin{aligned} \Vert u\Vert _{\varphi }=\inf \left\{ \lambda >0: \int _{\Omega }\varphi \left( x,\frac{|u(x)|}{\lambda }\right) \le 1\right\} . \end{aligned}$$

The functional

$$\begin{aligned} \rho _{\varphi }(u)=\int \limits _{\Omega }\varphi (x,|u(x)|)\,\textrm{d}x \end{aligned}$$

(3.1)

is a convex, left-continuous modular on $X_{\varphi }$ [9, 11, 24]. It is well known [9] that $L^{\varphi }(\Omega )$ is a Banach space. Since it will be needed in the sequel, we define the complementary function $\varphi ^*$ of $\varphi$ as

$$\begin{aligned}&\varphi ^*:\Omega \times [0,\infty )\longrightarrow [0,\infty )\end{aligned}$$

(3.2)

$$\begin{aligned}&\varphi ^*(x,t)=\sup \limits _{u\ge 0}\left( tu-\varphi (x,u)\right) . \end{aligned}$$

(3.3)

The complementary function $\varphi ^*$ is itself a MO-function (see [9]) and Hölder’s inequality holds, namely for $f\in L^{\varphi }(\Omega )$ and $g\in L^{\varphi ^*}(\Omega )$,

$$\begin{aligned} \int \limits _{\Omega }f(x)g(x)\,\textrm{d}x\le 2 \Vert f\Vert _{\varphi }\Vert g\Vert _{\varphi ^*}. \end{aligned}$$

(3.4)

If in addition

$$\begin{aligned} \int \limits _{K}\varphi (x,t)\,\textrm{d}x<\infty \end{aligned}$$

(3.5)

for any $K\subset \Omega$ with Lebesgue measure $|K|<\infty$ and

$$\begin{aligned} \inf \limits _{x\in \Omega }\varphi (x,1)>0, \end{aligned}$$

(3.6)

the Musielak–Orlicz Sobolev space $W^{1,\varphi }(\Omega )$ consisting of all functions in $L^{\varphi }(\Omega )$ whose distributional derivatives are in $L^{\varphi }(\Omega )$, is a Banach space when furnished with the norm

$$\begin{aligned} \Vert u\Vert _{1,\varphi }=\Vert u\Vert _{\varphi }+\Vert |\nabla u|\Vert _{\varphi }, \end{aligned}$$

where $\nabla$ stands for the gradient operator and $|\cdot |$ denotes the Euclidean norm in ${\mathbb R}^n$. The Sobolev space $W_0^{1,\varphi }(\Omega )$ is defined to be the closure of $C_0^{\infty }(\Omega )$ in $W^{1,\varphi }(\Omega )$.

4 Sobolev-type embeddings

The central idea of this Section is the Sobolev Embedding Theorem 4.7. In order to facilitate the flow of ideas we present a few definitions.

The Matuszewska index of an Orlicz function $\varphi$ was introduced by Matuszewska and Orlicz in [17].

Definition 4.1

For $\varphi$ as above and each $x\in \Omega$, set

$$\begin{aligned} M(x,t)=\limsup _{u\rightarrow \infty }\frac{\varphi (x,tu)}{\varphi (x,u)}. \end{aligned}$$

(4.1)

The Matuszewska index of $\varphi$ is defined to be

$$\begin{aligned} m(x)=\lim _{t\rightarrow \infty }\frac{\ln {M(x,t)}}{\ln {t}}= \inf _{t>1}\frac{\ln {M(x,t)}}{\ln {t}}. \end{aligned}$$

(4.2)

Definition 4.2

The limit (4.1) is said to be uniform if for each $\delta >0$ there exist $s_0>1$ and $T>1$ such that, for all $(x,t)\in \Omega \times [T,\infty )$, one as

$$\begin{aligned} M(x,t) -\delta<\frac{\varphi (x,ts_0)}{\varphi (x,s_0)}<M(x,t)+\delta . \end{aligned}$$

(4.3)

The following examples illustrate the above definition for some well known MO functions:

Example 4.3

Let $\Omega \subseteq {\mathbb R}^n$ be a bounded domain and

$$\begin{aligned} p:\Omega \longrightarrow (0,\infty ) \end{aligned}$$

be Borel-measurable. The MO function

$$\begin{aligned}&\varphi :\Omega \times [0,\infty )\longrightarrow [0,\infty ) \nonumber \\&\quad \varphi (x,t)=t^{p(x)} \end{aligned}$$

(4.4)

has Matuszewska index equal to p(x). In this case, the convergence (4.2) is trivially uniform on $\Omega$ and the limit (4.2) is clearly uniform.

Lemma 4.4

Let $\Omega \subset {\mathbb R}^n$ be a bounded domain and $\varphi$ an MO function as described above. If the Matuszewska index m is the restriction to $\Omega$ of a continuous function ${\tilde{m}}$ on the closure of $\Omega$, i.e.,

$$\begin{aligned} {\tilde{m}}: {\overline{\Omega }}\longrightarrow {\mathbb R}, \end{aligned}$$

(4.5)

and the convergence to the limits (4.1) and (4.2) is uniform, then there exist $C>1$, $T_0>1$ and $S_0>1$ such that uniformly in $\Omega$ it holds

$$\begin{aligned} \varphi (x,sT_0)\le C \varphi (x,s) \end{aligned}$$

(4.6)

for any $s\ge S_0$.

Condition (4.6) will be referred to as the $\Delta$ condition.

Proof

Fix $\delta >0$, then for some $T_0>1$ one has for any $t\ge T_0$, by virtue of (4.2)

$$\begin{aligned} t^{m(x)-\delta }< M(x,t)<t^{m(x)+\delta } \end{aligned}$$

(4.7)

uniformly in $\Omega$. By definition of M(x, t) and on account of the uniformity assumption of the infimum (4.1), there exists a positive number N for which, uniformly for $t\ge T_0$ and $x\in \Omega$, it holds that

$$\begin{aligned} \sup _{s\ge N}\frac{\varphi (x,st)}{\varphi (x,s)}<t^{m(x)+\delta }. \end{aligned}$$

(4.8)

In particular, for all $s\ge N$:

$$\begin{aligned} \varphi (x,sT_0)\le T_0^{\sup _{\Omega }m(x)+\delta }\varphi (x,s). \end{aligned}$$

$\square$

Corollary 4.5

There exists $S_0>1$ and a constant $C>1$ such that

$$\begin{aligned} \varphi (x,2s)\le C \varphi (x,s) \end{aligned}$$

(4.9)

for any $x\in \Omega$, $s\ge S_0$.

Lemma 4.6

If the statement of corollary 4.5 holds, then $\rho$-convergence is equivalent to norm-convergence in $L^{\varphi }(\Omega )$.

Proof

It suffices to show that if $(u_j)$ $\rho _{\varphi }$-converges to 0 and converges a.e. to 0. then it converges to 0 in the topology of the norm. This will be automatically implied by the validity of the equality

$$\begin{aligned} \lim _{j\rightarrow \infty }\rho _{\varphi }(\lambda x_n)=0 \end{aligned}$$

(4.10)

for any $\lambda >0.$ It is obviously necessary to show (4.10) only for $\lambda >1$. Let $N=[\log _2\lambda ]+1\ge 1$. A simple argument shows that for $C,S_0$ as in Corollary 4.5

$$\begin{aligned} \varphi (x,\lambda |u_n(x)|)\le \sum \limits _1^{N-1}\varphi (x,2S_0)+ C^N\varphi (x,(\lambda /2^N)|u_n(x)|). \end{aligned}$$

Since the second term in the right-hand side tends a.e. to 0 as $n\rightarrow \infty$, it follows that

$$\begin{aligned} \rho _{\varphi }(\lambda u_n)=\int \limits _{\Omega }\varphi (x,\lambda |u_n(x)|)\textrm{d}x\rightarrow 0\,\,\,\text {as}\,\,\,n\rightarrow \infty . \end{aligned}$$

(4.11)

$\square$

We refer the reader to [16] for the proof of the following theorem.

Theorem 4.7

Let $\Omega \subset {\mathbb R}^n$ be a bounded domain and

$$\begin{aligned} \varphi :\Omega \times [0,\infty )\longrightarrow {\mathbb R} \end{aligned}$$

a Musielak–Orlicz function that satisfies condition (3.6) and for which the limits in (4.1) and (4.2) are uniform; assume that the Matuszewska index m is the restriction to $\Omega$ of a continuous function ${\tilde{m}}$ defined on the closure of $\Omega$, that

$$\begin{aligned} 1<m_-=:\inf _{\Omega }m, \end{aligned}$$

and that there exists a function

$$\begin{aligned} \beta :(0,\infty )\longrightarrow (0,\infty ) \end{aligned}$$

such that, uniformly in $\Omega$ and for $t>0$:

$$\begin{aligned} \varphi (x,t)\le \beta (t). \end{aligned}$$

(4.12)

Then the embedding

$$\begin{aligned} W_0^{1,\varphi }(\Omega )\hookrightarrow L^{\varphi }(\Omega ) \end{aligned}$$

(4.13)

is compact.

Corollary 4.8

For $\varphi$ satisfying the conditions of Theorem 4.7, there exists a positive constant C depending only on $n,\Omega , \varphi$ such that for any $u\in W^{1,\varphi }_0(\Omega )$

$$\begin{aligned} \Vert u\Vert _{\varphi }\le C\Vert |\nabla u|\Vert _{\varphi }. \end{aligned}$$

(4.14)

Proof

See [16]. $\square$

5 $\Delta _2$-type Musielak–Orlicz functions

From now on we assume that $\Omega \subseteq {\mathbb R}^n$ is a bounded domain satisfying the cone condition and $\varphi$ is a MO-function satisfying the conditions of Theorem 4.7. Denote the conjugate of $\varphi$ by $\varphi ^*$.

Theorem 5.1

Let $\varphi$ be a MO function on $\Omega$; assume that $\varphi$ satisfies the conditions of Theorem 4.7; in particular, $\varphi$ satisfies the $\Delta$ condition 4.6, i.e., for some $K>0$, $S_0>1$ it holds that

$$\begin{aligned} \varphi (x,2s)\le K\varphi (x,s)\,\,\,\text {for all}\,\,\,s\ge S_0,\,x\in \Omega . \end{aligned}$$

(5.1)

Then,

$$\begin{aligned} \sup \left\{ \rho _{\varphi }(u): \rho _{\varphi }(|\nabla u|)\le r\right\} <\infty . \end{aligned}$$

(5.2)

Proof

It follows from (5.1) that for arbitrary $v\in L^{\varphi }(\Omega )$ (recall that $\varphi$ is nonnegative and nondecreasing)

$$\begin{aligned} \rho _{\varphi }(2v)&=\int \limits _{\Omega }\varphi (x,2|v(x)|)\textrm{d}x\nonumber \\&=\left( \int \limits _{|v|<S_0}+\int \limits _{|v|\ge S_0}\right) \varphi (x,2|v(x)|)\textrm{d}x \nonumber \\&\le \left( \int \limits _{|v|<S_0}\varphi (x,2S_0)\textrm{d}x+ K\rho _{\varphi }(v)\right) \nonumber \\&\le \left( \int \limits _{\Omega }\varphi (x,2S_0)\textrm{d}x+ K\rho _{\varphi }(v)\right) . \end{aligned}$$

(5.3)

If $r\ge 1$ and $u\in W^{1,\varphi }_0(\Omega )$ with

$$\begin{aligned} \rho _{\varphi }(|\nabla u|)\le r, \end{aligned}$$

(5.4)

it is a simple matter to verify that if $\Vert |\nabla u|\Vert _{\varphi }\ge 1$

$$\begin{aligned} 1&=\rho _{\varphi }\left( \frac{|\nabla u|}{\Vert |\nabla u|\Vert _{\varphi }}\right) \le \frac{1}{\Vert |\nabla u|\Vert _{\varphi }}\rho _{\varphi }(|\nabla u|)\\&\le \frac{1}{\Vert |\nabla u|\Vert _{\varphi }} r; \end{aligned}$$

it is thus clear that if (5.4) holds, then:

$$\begin{aligned} \Vert |\nabla u|\Vert _{\varphi }\le r. \end{aligned}$$

Therefore,

$$\begin{aligned} \Vert |\nabla u|\Vert _{\varphi }\le \max \{1,r\}=b. \end{aligned}$$

It follows from the preceding reasoning in conjunction with Poincaré inequality that if $\rho _{\varphi }(|\nabla u|)\le r$, then, for some $C>0$,

$$\begin{aligned} \Vert u\Vert _{\varphi }\le C\Vert |\nabla u|\Vert _{\varphi }\le Cb. \end{aligned}$$

Hence,

$$\begin{aligned} \rho _{\varphi }\left( \frac{u}{Cb}\right) \le 1. \end{aligned}$$

(5.5)

If $Cb<2$, (5.5) implies

$$\begin{aligned} \rho _{\varphi }(u)&=\rho _{\varphi }\left( Cb\frac{u}{Cb}\right) \le \rho _{\varphi }\left( 2\frac{u}{Cb}\right) \nonumber \\ {}&\le \int \limits _{\Omega }\varphi (x,2S_0)\textrm{d}x+ K\rho _{\varphi }\left( \frac{u}{Cb}\right) \nonumber \\ {}&\le \int \limits _{\Omega }\varphi (x,2S_0)\textrm{d}x+K. \end{aligned}$$

(5.6)

Otherwise, on account of the iteration of inequality (5.3)

$$\begin{aligned} \rho _{\varphi }(u)&=\rho _{\varphi }\left( Cb\frac{u}{Cb}\right) \nonumber \\&\le \rho _{\varphi }\left( 2^{[\log _2{Cb}]+1}\frac{u}{Cb}\right) \nonumber \\ {}&\le \left( 1+K+...K^{[\log _2{Cb}]}\right) \int \limits _{\Omega }\varphi (x,2S_0)\textrm{d}x+K^{[\log _2{Cb}]+1}. \end{aligned}$$

(5.7)

In all, (5.6) and (5.7) yield (5.2). $\square$

An immediate consequence of the preceding theorem is the following functional-analytic result:

Lemma 5.2

For $\varphi$ as in Theorem 4.7 and $r>0$, the modular ball

$$\begin{aligned} B_r:=\left\{ u\in W^{1,\varphi }_0(\Omega ): \rho _{\varphi }(|\nabla u|) \le r\right\} \end{aligned}$$

is weakly closed.

Proof

$B_r$ is clearly convex. It suffices to show that it is also norm-closed. If $(u_j)$ norm-converges to $u\in W^{1,\varphi }_0(\Omega )$, then $\rho _{\varphi }(|\nabla (u_j-u)|)\rightarrow 0$ as $n\rightarrow \infty$ and there is no loss of generality in assuming that $\nabla u_j\rightarrow \nabla u$ a.e. in $\Omega$. Theorem 4.7 guarantees that $(u_j)$ can be chosen so that $u_j\rightarrow u$ a.e. in $\Omega$. Since

$$\begin{aligned} \rho _{\varphi }(|\nabla u|)=\int \limits _{\Omega }\varphi (x, \lim _{n\rightarrow \infty }|\nabla u_n(x)|)\textrm{d}x\le \liminf _{n\rightarrow \infty }\int \limits _{\Omega }\varphi (x,|\nabla u_n(x)|)\textrm{d}x\le r, \end{aligned}$$

(5.8)

it follows that $u\in B_r$, i.e., $B_r$ is norm-closed (and convex) and hence weakly closed. $\square$

6 Differentiability properties

Aiming at a full description of the Fréchet derivative of the functionals to be introduced momentarily, a further assumption is imposed unto the MO function $\varphi$ at this point, namely, it is from now on required that $\varphi$ be an N function. More precisely:

Definition 6.1

An MO function is said to be an N-function iff it satisfies the condition

$$\begin{aligned} \lim _{t\rightarrow 0}\frac{\varphi (x,t)}{t}=0 \,\,\,a.e.. \end{aligned}$$

(6.1)

It is well known [9] that if $\varphi$ is an N-function, it can be written as

$$\begin{aligned} \varphi (x,t)=\int _0^t \phi (x,s)\,ds, \end{aligned}$$

(6.2)

where $\phi (x,\cdot )$ is the right t-derivative of $\varphi$. On the other hand, the conjugate function $\varphi ^*$ can be written as

$$\begin{aligned} \varphi ^*(x,t)=\int _0^t\phi ^{-1}(x,s)\,ds. \end{aligned}$$

(6.3)

The proof of the following theorem can be found in [14, 22]; we include it here in the interest of completeness.

Theorem 6.2

Let $\varphi$ be an N-function; assume that

$$\begin{aligned} \varphi _t(x,t)=\frac{\partial \varphi }{\partial t}(x,t) \end{aligned}$$

is continuous $a.e.\,\,x\in \Omega$. Define the operator $T_{\varphi _t}$ as

$$\begin{aligned}&T_{\varphi _t}:{{\mathcal {M}}}(\Omega )\longrightarrow {{\mathcal {M}}}(\Omega )\\&T_{\varphi _t}(u)=\varphi _t(x,|u(x)|)=\frac{\partial \varphi }{\partial t}(x,|u(x)|). \end{aligned}$$

Then, from the assumption

$$\begin{aligned} T_{\varphi _t}(L^{\varphi }(\Omega ))\subseteq L^{\varphi ^*}(\Omega ) \end{aligned}$$

(6.4)

it follows that the operator

$$\begin{aligned} T_{\varphi _t}:L^{\varphi }(\Omega )\longrightarrow L^{\varphi ^*}(\Omega ) \end{aligned}$$

is continuous and bounded.

Proof

Assume $(u_n)\subseteq L^{\varphi }(\Omega )$ converges to $u\in L^{\varphi }(\Omega )$. On $\Omega \times [0,\infty )$ define

$$\begin{aligned} w(x,t)=\varphi _t (x,|u(x)+t|)-\varphi _t(x,|u(x)|); \end{aligned}$$

then on account of the assumption on $\varphi _t$, w is a Carathéodory function and $w(x,0)=0$. If

$$\begin{aligned} T_w:L^{\varphi }(\Omega )\longrightarrow L^{\varphi ^*}(\Omega ) \end{aligned}$$

is continuous at 0, then $T_w(u_n-u)\longrightarrow 0$ in $L^{\varphi ^*}(\Omega )$ as $n\rightarrow \infty .$ If $\varphi ^*$ satisfies the $\Delta _2$ condition, the latter is equivalent to

$$\begin{aligned} \rho _{\varphi ^*}(T_w(u_n-u))=\int \limits _{\Omega }\varphi ^*(x,\left| \varphi _t (x,|u_n(x)|)-\varphi _t(x,|u(x)|)\right| )\rightarrow 0\,\,\,\text {as}\,\,\,n\rightarrow \infty , \end{aligned}$$

that is

$$\begin{aligned} T_{\varphi _t}(u_n)\longrightarrow T_{\varphi _t}(u)\,\,\,\text {in}\,\,L^{\varphi ^*}(\Omega )\,\,\text {as}\,\,\,n\rightarrow \infty . \end{aligned}$$

Therefore, it is enough to show that $T_{\varphi _t}$ is continuous at 0 under the assumption that $\varphi _t(x,0)=0\,\,a.e.$ in $\Omega$. Assume that $T_{\varphi _t}$ is not continuous at 0; let $r>0$ and let $(u_n)$ be a sequence that converges to 0 in $L^{\varphi }(\Omega )$, for which

$$\begin{aligned} \Vert T_{\varphi _t}(u_n)\Vert _{\varphi ^*}\ge r\,\,\,\text {for any}\,\,n\in {\mathbb N}. \end{aligned}$$

Since norm convergence implies modular convergence, one can, without loss of generality assume that

$$\begin{aligned} \max \left\{ \rho _{\varphi }(u_n), \Vert u_n\Vert _{\varphi }\right\} <\frac{1}{2^n}, \end{aligned}$$

and hence that $\sum _{n=1}^{\infty }\int \limits _{\Omega }\varphi (x,|u_n(x)|)\,\textrm{d}x<\infty .$ Due to the validity of the $\Delta _2$ condition for $\varphi ^*$, norm convergence and modular convergence are equivalent on $L^{\varphi ^*}(\Omega )$. It follows that there exists $\epsilon (r)>0$ such that $\rho _{\varphi ^*}\left( T_{\varphi _t}(u_n)\right) \ge \epsilon (r)$ for any $n\in {\mathbb N}.$ We next claim the existence of a sequence of real numbers $(\epsilon _k)$, a sequence $(\Omega _k)$ of subsets of $\Omega$ and a subsequence $(u_{n_k})$ of $(u_n)$ satisfying the following conditions:

(i):

$\epsilon _{k+1}<\frac{1}{2}\epsilon _k$,

(ii):

$|\Omega _k|\le \epsilon _k$,

(iii):

$\int \limits _{\Omega _k}\varphi ^*(x,|T_{\varphi _t}(u_{n_k})|)\,\textrm{d}x>\frac{2}{3}\epsilon (r)$.

(iv):

If $E\subseteq \Omega$ is measurable and $|E|<2\epsilon _{k+1}$, then

$$\begin{aligned} \int \limits _{E}\varphi ^*(x,|T_{\varphi _t}(u_{n_k})|)\,\textrm{d}x<\frac{\epsilon (r)}{3}. \end{aligned}$$

Set $\Omega _1=\Omega ,$ $\epsilon _1=|\Omega |$, $n_1=1.$ We assume that $\epsilon _k$, $n_k$ and $\Omega _k$ are given, then, by assumption, $\varphi _t(\cdot ,|u_{n_k}(\cdot )|)\in L^{\varphi ^*}(\Omega )$ and on account of the $\Delta _2$ condition one has

$$\begin{aligned} \int \limits _{\Omega }\varphi ^*(x, |\varphi _t(x,|u_{n_k}(x)|)|)\,\textrm{d}x<\infty . \end{aligned}$$

Since the measure

$$\begin{aligned} A\longrightarrow \mu (A)=\int \limits _{A}\varphi ^*(x, |\varphi _t(x,|u_{n_k}(x)|)|)\,\textrm{d}x \end{aligned}$$

defined on the Borel $\sigma$ algebra ${{\mathcal {B}}}$ of subsets of $\Omega$ is absolutely continuous with respect to the Lebesgue measure, one can find $\epsilon _{k+1}$ such that any $X\in {{\mathcal {B}}}$ with $|X|<2\epsilon _{k+1}$ satisfies $\int \limits _{X}\varphi ^*(x, |\varphi _t(x,|u_{n_k}(x)|)|)\,\textrm{d}x< \frac{\epsilon (r)}{3}.$ The assumption $\epsilon _k \le 2\epsilon _{k+1}$ would contradict (iii). We now proceed to the construction of $\Omega _{k+1}$ and $n_{k+1}$.

It is well known [14] that the strong convergence of $(u_n)$ in $L^{\varphi }(\Omega )$ implies the convergence in measure of both $(T_{\varphi _t}(u_n))$ and $(T_{\varphi ^*}(T_{\varphi _t}(u_n)))$. Consequently, there exists $n_{k+1}\in {\mathbb N}$ such that

$$\begin{aligned} \left| \left\{ x\in \Omega : \left| T_{\varphi ^*}(T_{\varphi _t}(u_{n_{k+1}}))\right| >\frac{\epsilon (r)}{3|\Omega |} \right\} \right|<\epsilon _{k+1}<\frac{\epsilon _k}{2}<\epsilon _k. \end{aligned}$$

Define

$$\begin{aligned} \Omega _{k+1}=\left\{ x\in \Omega : \left| T_{\varphi ^*}(T_{\varphi _t}(u_{n_{k+1}}))\right| >\frac{\epsilon (r)}{3|\Omega |} \right\} . \end{aligned}$$

Next,

$$\begin{aligned} \int \limits _{\Omega _{k+1}}\varphi ^*(x,|T_{\varphi _t}(u_{n_{k+1}})|)\,\textrm{d}x&=\left( \int \limits _{\Omega } -\int \limits _{\Omega \setminus \Omega _{k+1}}\right) \varphi ^*(x,|T_{\varphi _t}(u_{n_{k+1}})|)\,\textrm{d}x\\&>\epsilon (r)-\frac{\epsilon (r)}{3}=\frac{2\epsilon (r)}{3}. \end{aligned}$$

By construction

$$\begin{aligned} \left| \bigcup \limits _{j=k+1}^{\infty }\Omega _j\right| \le \sum _{j=k+1}^{\infty } \epsilon _j<2\epsilon _{k+1}. \end{aligned}$$

(6.5)

Set

$$\begin{aligned} v(x)={\left\{ \begin{array}{ll} u_{n_k}(x)\,\,\,\,\text {if}\,\,\,\,x\in \Omega _k\setminus \bigcup \limits _{j=k+1}^{\infty }\Omega _j \\ 0\,\,\,\,\text {otherwise}. \end{array}\right. } \end{aligned}$$

It is clear that $v\in L^{\varphi }(\Omega )$. Next, observe that:

$$\begin{aligned} \int \limits _{\Omega }\varphi ^*(x,T_{\varphi _t}(v(x)))\,\textrm{d}x&\ge \sum _{k=1}^{\infty }\int \limits _{\Omega _k\setminus \cup _{j=k+1}^{\infty }\Omega _j}\varphi ^*(x,T_{\varphi _t}(u_{n_k}(x)))\,\textrm{d}x \\ {}&=\sum _{k=1}^{\infty }\left( \int \limits _{\Omega _k}-\int \limits _{ \cup _{j=k+1}^{\infty }\Omega _j}\right) \varphi ^*(x,T_{\varphi _t}(u_{n_k}(x)))\,\textrm{d}x\\ {}&\ge \sum _{k=1}^{\infty }\left( \frac{2}{3}\epsilon (r)-\frac{1}{3}\epsilon (r)\right) \\ {}&=\infty , \end{aligned}$$

which follows by (6.5) and condition (iv). This contradicts assumption (6.4). Hence, $T_{\varphi _t}$ is continuous at 0. $\square$

We next prepare the ground for the next lemma, which deals with the differentiability properties needed in the sequel. Let $M:{\mathbb R}\longrightarrow [0,\infty )$ be continuous and write

$$\begin{aligned} {\hat{M}}(s)=\left( \int _0^{s}M(t)\,dt\right) . \end{aligned}$$

(6.6)

Consider the maps

$$\begin{aligned}&F:L^{\varphi }(\Omega )\longrightarrow [0,\infty ) \nonumber \\&\quad F(u)=\rho _{\varphi }(u) \end{aligned}$$

(6.7)

and

$$\begin{aligned}&H:W_0^{1,\varphi }(\Omega )\longrightarrow [0,\infty )\nonumber \\&\quad H(u)={\hat{M}}\left( \rho _{\varphi }(|\nabla u|)\right) . \end{aligned}$$

(6.8)

Recall that a MO function $\varphi$ is said to be locally integrable if for any $t>0$ and any subset ${{\mathcal {W}}}\subseteq \Omega$ with $\mu ({{\mathcal {W}}})<\infty$ one has

$$\begin{aligned} \int \limits _{{{\mathcal {W}}}}\varphi (x,t)\,\textrm{d}x<\infty . \end{aligned}$$

Lemma 6.3

In the terminology of the preceding paragraph, let $\varphi$ be an N function; suppose that $\frac{\partial }{\partial t}\varphi (x,\cdot )$ is continuous a.e. x. Assume that the complementary function $\varphi ^*$ of $\varphi$ satisfies the $\Delta$ condition and is locally integrable. If the maps

$$\begin{aligned}&D_1:L^{\varphi }(\Omega )\longrightarrow L^{\varphi ^{*}}(\Omega )\end{aligned}$$

(6.9)

$$\begin{aligned}&D_1(u)=\frac{\partial }{\partial t}\varphi (\cdot ,|u(\cdot )|) \end{aligned}$$

(6.10)

and

$$\begin{aligned}&D_2:W_0^{1,\varphi }(\Omega )\longrightarrow L^{\varphi ^{*}}(\Omega ) \end{aligned}$$

(6.11)

$$\begin{aligned}&D_2(u)=\frac{\partial }{\partial t}\varphi (\cdot ,|\nabla u(\cdot )|) \end{aligned}$$

(6.12)

are well defined, then the functionals (6.7) and (6.8) are Fréchet differentiable for $u\ne 0$. In this case, the derivatives of F and H at $u\ne 0$ are given, respectively by

$$\begin{aligned} \langle F^{'}(u),h\rangle =\int \limits _{\Omega }\frac{\partial \varphi }{\partial s}(x,|u(x)|)\frac{u(x)}{|u(x)|}h(x)\,\textrm{d}x, \end{aligned}$$

(6.13)

and by

$$\begin{aligned} \langle H^{'}(u),h\rangle =\langle -M\left( \rho _{\varphi }(|\nabla u)\right) \text {div}\left( \frac{\partial \varphi }{\partial s}(x,|\nabla u(x)|)\frac{\nabla u(x)}{|\nabla u(x)|},h\rangle \right) , \end{aligned}$$

(6.14)

with the understanding that $\frac{x}{|x|}=0$ if $x=0$.

Proof

It suffices to show that under the stipulated conditions (6.13) holds; a straightforward application of the chain rule will yield the full statement of the Lemma.

Observe that for $u(x)\ne 0$, $h\in L^{\varphi }(\Omega )$ one has

$$\begin{aligned}{} & {} \frac{|u(x)+th(x)|-|u(x)|}{t}=\frac{2u(x)h(x)+t(h(x))^2}{|u(x)+th(x)|+|u(x)|}\\{} & {} \quad \longrightarrow \frac{u(x)h(x)}{|u(x)|}\,\,\,\text {as}\,\,\,t\longrightarrow 0. \end{aligned}$$

Therefore, for some $\theta \in (|u(x)|,|u(x)+th(x)|)$,

$$\begin{aligned}{} & {} \frac{\varphi (x,|u(x)+th(x)|)-\varphi (x,|u(x)|)}{t}=\frac{|u(x)+th(x)|-|u(x)|}{t}\frac{\partial \varphi }{\partial t}(x,\theta ) \\{} & {} \quad \longrightarrow \frac{u(x)h(x)}{|u(x)|} \frac{\partial \varphi }{\partial t}(x,|u(x)|)\,\,\,\text {as}\,\,\,t\longrightarrow 0. \end{aligned}$$

By hypothesis,

$$\begin{aligned} \int \limits _{\Omega }\left| h(x)\frac{\partial \varphi }{\partial t}(x,|u(x)|)\right| \,\textrm{d}x<\infty , \end{aligned}$$

(6.15)

which in conjunction with the above limit yields:

$$\begin{aligned} \lim _{t\longrightarrow 0}\int \limits _{\{x:u(x)\ne 0\}}\frac{|u(x)+th(x)|-|u(x)|}{t}\frac{\partial \varphi }{\partial t}(x,\theta (x))\textrm{d}x\nonumber \\=\int \limits _{\{x:u(x)\ne 0\}}\frac{u(x)h(x)}{|u(x)|} \frac{\partial \varphi }{\partial t}(x,|u(x)|)\textrm{d}x. \end{aligned}$$

(6.16)

On the other hand, one has by assumption, one has a.e. $x\in \Omega$:

$$\begin{aligned} \frac{\varphi (x,|th(x)|)}{t}\longrightarrow 0\,\,\, \text {as}\,\,\,t\longrightarrow 0 \end{aligned}$$

and by convexity, for $0<t<1$,

$$\begin{aligned} \frac{\varphi (x,|th(x)|)}{t}\le \varphi (x,|h(x)|). \end{aligned}$$

Since $h\in L^{\varphi }(\Omega )$, $\int _{\Omega }\varphi (x,|h(x)|)\,\textrm{d}x<\infty$, it is easily derived by way of application of Lebesgue’s theorem that

$$\begin{aligned}{} & {} \int \limits _{\{x:u(x)=0\}}\left| \frac{\varphi (x,|u(x)+th(x)|-\varphi (x,|u(x)|)}{t}\right| \,\textrm{d}x \nonumber \\{} & {} \quad =\int \limits _{\{x:u(x)=0\}}\varphi (x,t|h(x)|)/t\,\textrm{d}x\rightarrow 0\,\,\,\text {as}\,\,\,t\rightarrow 0. \end{aligned}$$

(6.17)

In all,

$$\begin{aligned} \frac{F(u+th)-F(u)}{t}= & {} \left( \int \limits _{\{x:u(x)\ne 0\}}+\int \limits _{\{x:u(x)=0\}}\right) \frac{\varphi (x,|u(x)+th(x)|)-\varphi (x,|u(x)|)}{t}\textrm{d}x \\{} & {} \rightarrow \int \limits _{\{x:u(x)\ne 0\}}\frac{u(x)h(x)}{|u(x)|} \frac{\partial \varphi }{\partial t}(x,|u(x)|)\textrm{d}x\,\,\,\text {as}\,\,\,t\longrightarrow 0. \end{aligned}$$

We conclude that F is Gâteaux differentiable and that its derivative is equal to the right-hand side of (6.13). The proof of the Gâteaux differentiability of G follows along the same lines. Hence, it suffices to prove that under the additional assumptions (6.9) and (6.11), the operators

$$\begin{aligned} {{\mathcal {L}}_F}&:L^{\varphi }(\Omega )\longrightarrow \left( L^{\varphi }(\Omega )\right) ^* \nonumber \\&{{\mathcal {L}}_f}(u)=F^{'}(u) \end{aligned}$$

(6.18)

$$\begin{aligned} {{\mathcal {L}}_G}&:W_0^{1,\varphi }(\Omega )\longrightarrow \left( W_0^{1,\varphi }(\Omega )\right) ^* \nonumber \\&{{\mathcal {L}}}(u)=G^{'}(u) \end{aligned}$$

(6.19)

are continuous at $u\ne 0$. A standard functional-analytic result guarantees that in this case F and G are Fréchet differentiable and that the Fréchet and the Gâteaux derivatives coincide. To this end, consider a convergent sequence $(u_j)\subset L^{\varphi }(\Omega )$, say $u_j\longrightarrow u$ in $L^{\varphi }(\Omega )$ as $j\longrightarrow \infty$: it is well known that there is no loss of generality by assuming that $u_j$ converges to u almost everywhere in $\Omega$ (see [9, 11])).

Since $\frac{\partial \varphi }{\partial t}(x,|u(x)|)\in L^{\varphi ^*}(\Omega )$ there exists $\lambda _0>0$ for which

$$\begin{aligned} \int \limits _{\Omega }\varphi ^*\left( x,\lambda _0\left| \frac{\partial \varphi }{\partial t}(x,|u(x)|)\right| \right) \textrm{d}x<\infty . \end{aligned}$$

(6.20)

Notice that for $\Vert h\Vert _{\varphi }\le 1$ one has

$$\begin{aligned}&\int \limits _{\{x:u_j(x)\ne 0,u(x)\ne 0\}}\left( \frac{u_j(x)}{|u_j(x)|}\frac{\partial \varphi }{\partial t}(x,|u_j(x)|)-\frac{u(x)}{|u(x)|}\frac{\partial \varphi }{\partial t}(x,|u(x)|))\right) h(x)\,\textrm{d}x \end{aligned}$$

(6.21)

$$\begin{aligned}&= \int \limits _{\{x:u_j(x)\ne 0,u(x)\ne 0\}}\frac{u_j(x)}{|u_j(x)|}\left( \frac{\partial \varphi }{\partial t}(x,|u_j(x)|)-\frac{\partial \varphi }{\partial t}(x,|u(x)|)\right) h(x)\,\textrm{d}x\nonumber \\&+ \int \limits _{\{x:u_j(x)\ne 0,u(x)\ne 0\}}\left( \frac{u_j(x)}{|u_j(x)|}-\frac{u(x)}{|u(x)|}\right) \frac{\partial \varphi }{\partial t}(x,|u(x)|)h(x)\,\textrm{d}x. \end{aligned}$$

(6.22)

Theorem 6.2 guarantees the continuity of the map (6.9); it is apparent from this fact in conjunction with Hölder’s inequality (3.4) that the integral in (6.21) tends to 0 as j tends to infinity. As to the remaining integral, set, for $j\in {\mathbb N}$:

$$\begin{aligned} \left| \frac{\partial \varphi }{\partial t}(x,|u(x)|)\right|&=f(x)\\ r_j(x)&=\left( \frac{u_j(x)}{|uj(x)|}-\frac{u(x)}{|u(x)|}\right) f(x). \end{aligned}$$

Recall that in the terminology of Lemma 4.4,

$$\begin{aligned} \varphi ^{*}(x,sT_0)\le C\varphi ^{*}(x,s), \end{aligned}$$

(6.23)

for any $s\ge S_0$. For $\lambda _0$ as in (6.20) and any $\lambda >2\lambda _0$ let k be defined by the inequalities

$$\begin{aligned} 2^{k-2}<\lambda /\lambda _0\le 2^{k-1}. \end{aligned}$$

Notice that $|r_j|\le 2$. For $S_0$ as in Lemma 4.4, one readily obtains, for any positive integer m, using the monotonicity of $\varphi ^{*}$ in the second variable

$$\begin{aligned}&\varphi ^*\left( x,2^m\lambda _0f(x)\right) \nonumber \\&\quad =\varphi ^*\left( x,2^m\lambda _0f(x)\right) \left( I_{[S_0,\infty )}\left( 2^{m-1}\lambda _0f(x)\right) +I_{[0,S_0)}\left( 2^{m-1}\lambda _0f(x)\right) \right) \nonumber \\&\quad \le C \varphi ^{*} (x, 2^{m-1}\lambda _0f(x))+\varphi ^{*}(x,2S_0). \end{aligned}$$

(6.24)

The iteration of the preceding inequality yields

$$\begin{aligned} \varphi ^*\left( x,\lambda r_j\right)&\le \varphi ^*\left( x,2^k\lambda _0f(x)\right) \\&=\varphi ^*\left( x,2^k\lambda _0f(x)\right) \left( I_{[S_0,\infty )}\left( 2^{k-1}\lambda _0f(x)\right) +I_{[0,S_0)}\left( 2^{k-1}\lambda _0f(x)\right) \right) \\&\le \left( \sum _{i=0}^{k-1} C^i\right) \varphi ^*(x,2S_0) +C^k\varphi ^*(x,\lambda _0f(x)). \end{aligned}$$

A routine application of Lebesgue’s dominated convergence quickly shows that

$$\begin{aligned} \int \limits _{\{x:u_j(x)\ne 0,u(x)\ne 0\}}\varphi ^*\left( x,\lambda \left( \frac{u_j(x)}{|u_j(x)|}-\frac{u(x)}{|u(x)|}\right) \frac{\partial \varphi }{\partial t}(x,|u(x)|)\right) \textrm{d}x\longrightarrow 0\,\,\,\text {as}\,\,\,j\longrightarrow \infty . \end{aligned}$$

(6.25)

A similar argument shows that (6.25) holds for $0<\lambda \le 2\lambda _0$. It follows from the arbitrariness of $\lambda$ that

$$\begin{aligned} \left\| \left( \frac{u_j}{|u_j|}-\frac{u}{|u|}\right) \frac{\partial \varphi }{\partial t}(\cdot ,|u|)\right\| _{L^{\varphi ^*}(\Omega )}\rightarrow 0 \,\,\,\text {as}\,\,\,j\rightarrow \infty . \end{aligned}$$

On account of Hölders inequality (3.4), the integral (6.22) is bounded by

$$\begin{aligned} 2\left\| \left( \frac{u_j}{|u_j|}-\frac{u}{|u|}\right) \frac{\partial \varphi }{\partial t}(\cdot ,|u|)\right\| _{L^{\varphi ^*}(\Omega )}\longrightarrow 0\,\,\,\text {as}\,\,\,j\longrightarrow \infty . \end{aligned}$$

(6.26)

The proof of the continuity of ${{\mathcal {L}}}_G$ follows along the same lines and will be skipped.

This concludes the continuity argument and hence F and G are Fréchet differentiable. $\square$

Lemma 6.4

If

$$\begin{aligned} {\hat{M}}:[0,\infty )\longrightarrow [0,\infty ) \end{aligned}$$

is strictly increasing, the modular ${\hat{M}}$- ball

$$\begin{aligned} B_r= \left\{ u\in W^{1,\varphi }_0(\Omega ): {\hat{M}}\rho _{\varphi }(|\nabla u|)\le r\right\} \end{aligned}$$

(6.27)

is weakly closed in $W^{1,\varphi }_0(\Omega )$.

Proof

For any $r>0$ set $s_r={\hat{M}}^{-1}(r)$; it is then clear from the above assumptions that

$$\begin{aligned} B_r=V_{s_r} =\left\{ u\in W^{1,\varphi }_0(\Omega ): \rho _{\varphi }(|\nabla u|)\le s_r\right\} \end{aligned}$$

(6.28)

and the latter set is weakly closed (Lemma 5.2). $\square$

7 Kirchhoff-type eigenvalue problem

We are now ready to prove the main result of this work.

Theorem 7.1

Let $M\in C\left( (0,\infty ), (0,\infty )\right)$. Set ${\hat{M}}(t)=\int _0^tM(s)ds.$ Then, for any $r>0$, there exists a solution $(u,\lambda )\in W_0^{1,\varphi }(\Omega )\times (0,\infty )$ to the equation

$$\begin{aligned} -M\left( \rho _{\varphi }(|\nabla u|)\right) \text {div}\left( \frac{\partial \varphi }{\partial s}(x,|\nabla u(x)|)\frac{\nabla u(x)}{|\nabla u(x)|}\right) =\lambda \frac{\partial \varphi }{\partial s}(x,|u(x)|)\frac{u(x)}{|u(x)|}, \end{aligned}$$

(7.1)

satisfying

$$\begin{aligned} {\hat{M}}(\rho _{\varphi }(|\nabla u|))=r \end{aligned}$$

(7.2)

Proof

Theorem 5.1 guarantees that for any $r>0$,

$$\begin{aligned} 0<\sup \left\{ \rho _{\varphi }\left( u\right) :u\in B_r \right\} =S_r<\infty . \end{aligned}$$

(7.3)

We next observe that Theorem 4.7 implies the existence of a sequence $(u_j)\subset V_{s_r}$ with $u_j\rightharpoonup u_0$ in $W^{1,\varphi }_0(\Omega )$ and $u_j\longrightarrow u_0$ in $L^{\varphi }(\Omega )$ such that

$$\begin{aligned} \rho _{\varphi }\left( u\right) \longrightarrow S_r=\rho _{\varphi }\left( u_0\right) . \end{aligned}$$

(7.4)

To see this, we notice that a.e. in $\Omega$,

$$\begin{aligned} \varphi (x,|u_n(x)|)\longrightarrow \varphi (x,|u(x)|) \end{aligned}$$

and that on account of convexity, for any $n\in {\mathbb N}$:

$$\begin{aligned} \varphi (x,|u_n(x)|)&\le \frac{1}{2}\varphi (x,2|u_n(x)-u(x)|)+\frac{1}{2}\varphi (x,2|u(x)|). \end{aligned}$$

(7.5)

Select n large enough so that $2\Vert u-u_n\Vert _{\varphi }<1$; for such n, it holds, by way of the convexity of $\varphi (x,\cdot )$,

$$\begin{aligned} \int \limits _{\Omega }\frac{1}{2}\varphi (x,2|u_n(x)-u(x)|)\textrm{d}x&= \int \limits _{\Omega }\frac{1}{2}\varphi \left( x,\frac{2|u_n(x)-u(x)|2\Vert u-u_n\Vert _{\varphi }}{2\Vert u-u_n\Vert _{\varphi }}\right) \textrm{d}x \nonumber \\&\le \Vert u-u_n\Vert _{\varphi } \int \limits _{\Omega }\varphi \left( x,\frac{|u_n(x)-u(x)|}{\Vert u-u_n\Vert _{\varphi }}\right) \textrm{d}x \end{aligned}$$

(7.6)

Denote the left-hand side and the right-hand side of (7.5) by $v_n$ and $w_n$ respectively. Then the following conditions hold:

(i):

$v_n(x)\rightarrow v(x)=\varphi (x,|u(x)|)\in L^1(\Omega )$ a.e. in $\Omega$

(ii):

$w_n(x)\rightarrow w(x)=\frac{1}{2}\varphi (x,2|u(x)|)\in L^1(\Omega )$ a.e. in $\Omega$

(iii):

$v_n, w_n\in L^{1}(\Omega )$ for any $n\in {\mathbb N}$

(iv):

$\int \limits _{\Omega }w_ndx\rightarrow \int \limits _{\Omega }\frac{1}{2}\varphi (x,2|u|)\textrm{d}x=\frac{1}{2}\rho _{\varphi }(2u).$

Since $w-v\ge 0$ a.e in $\Omega$, Fatou’s Lemma leads to:

$$\begin{aligned} \int \limits _{\Omega }(w-v)\textrm{d}x&\le \int \limits _{\Omega }w\,\textrm{d}x+\liminf \limits _n\int \limits _{\Omega }(-v_n)\,\textrm{d}x \\&=\int \limits _{\Omega }w\,\textrm{d}x-\limsup \limits _n\int \limits _{\Omega }v_n\,\textrm{d}x \end{aligned}$$

and

$$\begin{aligned} \int \limits _{\Omega }(w+v)\textrm{d}x&\le \int \limits _{\Omega }w\,\textrm{d}x+\liminf \limits _n\int \limits _{\Omega }v_n\,\textrm{d}x. \end{aligned}$$

The two last statements yield

$$\begin{aligned} \lim _{n\rightarrow \infty }\int \limits _{\Omega }\varphi (x,|u_n(x)|)\textrm{d}x=\int \limits _{\Omega }\varphi (x,|u(x)|)\textrm{d}x \end{aligned}$$

or, equivalently

$$\begin{aligned} \rho _{\varphi }(u_n)\longrightarrow \rho _{\varphi }(u)\,\,\text {as}\,\,n\rightarrow \infty . \end{aligned}$$

(7.7)

By construction $\rho _{\varphi }(u_n)\longrightarrow S_r$; (7.7) is therefore the desired result.

Lemma 6.4 yields ${\hat{M}}\left( \rho _{\varphi }(|\nabla u_0|)\right) \le r.$ Furthermore, the continuity and monotonicity of the modular $\rho _{\varphi }$ immediately yield

$$\begin{aligned} {\hat{M}}(\rho _{\varphi }(|\nabla u_0|))=r. \end{aligned}$$

As is apparent from the above, $u_0$ is a solution of the constrained maximization problem of the type

$$\begin{aligned} \max F\left( v\right) \,\,\,with\,\,\,G(v)=r. \end{aligned}$$

(7.8)

It is a routine matter to show, via the Implicit Function Theorem, that the preceding statement implies that

$$\begin{aligned} \text {ker}G^{\prime }(u_0)=\text {ker}F^{\prime }(u_0), \end{aligned}$$

(7.9)

from which it is clear (since neither functional is null) that $u_0$ satisfies equation (7.1) for some $\lambda >0$. This concludes the proof of the claim. $\square$

8 Applications

A particular instance of Theorem 7.1 deserves to be stressed, namely its implication in the consideration of variable exponent Lebesgue spaces.

More precisely, if $\Omega \subset {\mathbb R}^n$ is bounded then

$$\begin{aligned}&\varphi :\Omega \times [0,\infty )\longrightarrow [0,\infty ) \end{aligned}$$

(8.1)

$$\begin{aligned}&\varphi (x,t)=\frac{t^{p(x)}}{p(x)} \end{aligned}$$

(8.2)

satisfies the conditions of Theorem 4.7 iff p is the restriction to $\Omega$ of a function ${\tilde{p}}\in C({\mathbb R}^n,{\mathbb R})$ and

$$\begin{aligned} 1<p_-=\inf \limits _{x\in \Omega }p(x)\le p_+=\sup \limits _{x\in \Omega }p(x)<\infty . \end{aligned}$$

(8.3)

The conjugate function is clearly given by

$$\begin{aligned} \varphi ^*(x,t)=\frac{p(x)-1}{p(x)}t^{\frac{p(x)}{p(x)-1}}. \end{aligned}$$

(8.4)

It is straightforward to verify the conditions of Lemma 6.3 for this case. It is customary to write, in this case

$$\begin{aligned} \rho _p(u)=\int \limits _{\Omega }\frac{|u(x)|^{p(x)}}{p(x)}\textrm{d}x. \end{aligned}$$

(8.5)

Therefore, Theorem 7.1 yields the following result:

Theorem 8.1

If $\Omega \subset {\mathbb R}^n$ is bounded, M satisfies the conditions of Theorem 7.1 and

$$\begin{aligned} p:\Omega \longrightarrow (1,\infty ) \end{aligned}$$

is a variable exponent satisfying the assumptions (8.3) then for each $r>0$ there exists a solution $(u_0,\lambda )$ to the eigenvalue problem

$$\begin{aligned} M\left( \rho _{p}(|\nabla u|)\right) div\left( |\nabla u|^{p(x)-2}\nabla u\right) =\lambda |u|^{p(x)-2}u \end{aligned}$$

(8.6)

with $M\left( \rho _{p}(|\nabla u_0|)\right) =r$.

With the aid of the following compactness theorem, the techniques used in Sections 6 and 7, one can derive Theorems 8.3 and 8.4, particular cases of which were obtained in [2] and [4], respectively, via different methods.

Theorem 8.2

[9, 15] Let $\Omega \subset {\mathbb R}^n$, $n>1$ be a bounded domain, $p\in C({\overline{\Omega }})$ with

$$\begin{aligned} 1<p_-\le p_+<n. \end{aligned}$$

(8.7)

For $0<\varepsilon <\frac{1}{n-1}$ and $q\in {{\mathcal {P}}}(\Omega )$ such that

$$\begin{aligned} q(x)<\frac{np(x)}{n-p(x)}-\epsilon , \end{aligned}$$

(8.8)

the embedding

$$\begin{aligned} W_0^{1,p(\cdot )}(\Omega )\hookrightarrow L^{q(\cdot )}(\Omega ) \end{aligned}$$

(8.9)

is compact.

Theorem 8.3

Given a function $M\in C\left( (0,\infty ), [0,\infty )\right)$, $M(t)>0$ for $t>0$. Under the hypotheses of Theorem 8.2, for any $r>0$, there exists a solution $(u,\lambda )\in W_0^{1,p(\cdot )}(\Omega )\times (0,\infty )$ to the equation

$$\begin{aligned} -M\left( \rho _{p}\left( |\nabla u|\right) \right) \text {div}\left( |\nabla u|^{p-2}\nabla u\right) =\lambda |u|^{q-2}u, \end{aligned}$$

(8.10)

satisfying

$$\begin{aligned} M\left( \rho _{p}\left( |\nabla u|\right) \right) =r. \end{aligned}$$

(8.11)

Proof

The proof follows from Theorem 7.1 and Theorem 8.3 by considering $\varphi (x,t)=\frac{t^{p(x)}}{p(x)}$ and by way of the bound

$$\begin{aligned} \sup \left\{ \int \limits _{\Omega }\frac{|u(x)|}{q(x)}: \rho _{p}(|\nabla u|)\le r\right\} <\infty , \end{aligned}$$

(8.12)

which is easy derived as in Theorem 5.1 via the compactness result of Theorem 8.2. $\square$

Theorem 8.4

Under the assumptions of Theorem 8.2, for any $0\le V\in L^{\infty }(\Omega )$ and $r>0$, there exists a solution $(u,\lambda )\in W_0^{1,p(\cdot )}(\Omega )\times (0,\infty )$ to the equation

$$\begin{aligned} -M\left( \rho _{p}\left( \frac{|\nabla u|}{p}\right) \right) \text {div}\left( |\nabla u|^{p-2}\nabla u\right) =\lambda V|u|^{q-2}u, \end{aligned}$$

(8.13)

satisfying

$$\begin{aligned} {\hat{M}}\left( \rho _{p}\left( \frac{|\nabla u|}{p}\right) \right) =r. \end{aligned}$$

(8.14)

Proof

The proof follows along the same lines as those of Theorem 7.1 by observing that the functional

$$\begin{aligned}L^{p(\cdot )}(\Omega ) \ni u\rightarrow T(u)=\int \limits _{\Omega }\frac{V(x)}{q(x)}|u(x)|^{q(x)}\textrm{d}x \end{aligned}$$

is Fréchet differentiable for $u\ne 0$ and that, for $h\in C_0^{\infty }(\Omega )$,

$$\begin{aligned}\langle T^{\prime }(u),h\rangle =\int \limits _{\Omega }V(x)|u(x)|^{q(x)-2}u(x)h(x)\textrm{d}x.\end{aligned}$$

If $B_r$, $r>0$, is defined as in Lemma 6.4, it follows as in the proof of Theorem 7.1 that there exists $u_0\in B_r$ with ${\hat{M}}\left( \int \limits _{\Omega }\left| \frac{\nabla u_0}{p(x)}\right| ^{p(x)}\textrm{d}x\right) =r$ such that

$$\begin{aligned}\int \limits _{\Omega }\frac{V(x)}{q(x)}|u_0(x)|^{q(x)}\textrm{d}x=\max \left\{ \int \limits _{\Omega }\frac{V(x)}{q(x)}|u(x)|^{q(x)}\textrm{d}x, u\in B_r\right\} .\end{aligned}$$

Reasoning mutatis mutandis as in the proof of Theorem 7.1 it can be shown that $u_0$ is in fact a sought-for solution to Problem 8.13. $\square$