Non-autonomous Eigenvalue Problems with Variable (p₁,p₂)-Growth

1 Introduction

The recent study of various nonlinear models described by partial differential equations with variable exponent is motivated by the rigorous mathematical description of many phenomena in applied sciences. In some cases the standard approach based on the theory of classical Lp and W1,p Lebesgue and Sobolev spaces is not adequate in the framework of material with non-homogeneities. For instance, electro-rheological fluids (sometimes referred to as “smart fluids”) or phenomena in image processing are described in a correct manner by mathematical models in which the exponent p is allowed to vary. This leads us to the study of variable exponents Lebesgue and Sobolev spaces, Lp⁢(x) and W1,p⁢(x), where p is a real-valued function. We refer to the work by Diening, Hästo, Harjulehto, and Ruzicka [6] for the abstract framework describing these spaces as well as to the monograph by Rădulescu and Repovš [15], which includes a thorough variational and topological analysis of several classes of problems with variable exponent (see also the survey paper by Rădulescu [14] and the papers by Colasuonno and Pucci [3] and Pucci and Zhang [13]).

We are interested in the study of a class of non-autonomous stationary problems, which are characterized by the fact that the associated energy density changes its ellipticity and growth properties according to the point. Problems of this type have been intensively studied starting with the pioneering contributions of Halsey [7] and Zhikov [17, 18, 19] in relationship with the analysis of the behaviour of strongly anisotropic materials in the context of the homogenization and nonlinear elasticity.

The study of non-homogeneous elliptic problems has been recently extended by Kim and Kim [8] to a new class of differential operators. Their contribution is a step forward in the analysis of nonlinear problems with variable exponent since it enables the understanding of problems with possible lack of uniform convexity. In the present paper, we extend this study to problems involving several non-homogeneous operators (as introduced in [8]) and we describe some spectral properties in relationship with two Rayleigh-type quotients. Section 2 includes some basic properties of function spaces with variable exponents. The main result is described in Section 3 while the proofs and some perspectives are presented in Section 4 of this paper.

2 Basic Properties of Spaces with Variable Exponent

Throughout this paper, we assume that Ω⊂ℝN is a bounded domain with smooth boundary.

Set

Assume that p∈C+⁢(Ω¯) and let

We define the Lebesgue space with variable exponent by

This function space is a Banach space if it is endowed with the norm

This norm is also called the Luxemburg norm. Then Lp⁢(x)⁢(Ω) is reflexive if and only if 1<p-≤p+<∞, and continuous functions with compact support are dense in Lp⁢(x)⁢(Ω) if p+<∞.

The standard inclusion between Lebesgue spaces generalizes to the framework of spaces with variable exponent, namely if 0<|Ω|<∞ and p1, p2 are variable exponents such that p1≤p2 in Ω, then there exists the continuous embedding Lp2⁢(x)⁢(Ω)↪Lp1⁢(x)⁢(Ω).

Let Lp′⁢(x)⁢(Ω) denote the conjugate space of Lp⁢(x)⁢(Ω), where 1/p⁢(x)+1/p′⁢(x)=1. Then for all u∈Lp⁢(x)⁢(Ω) and v∈Lp′⁢(x)⁢(Ω) the following Hölder-type inequality holds:

An important role in analytic arguments on Lebesgue spaces with variable exponent is played by the modular of Lp⁢(x)⁢(Ω), which is the map ρp⁢(x):Lp⁢(x)⁢(Ω)→ℝ defined by

If (un),u∈Lp⁢(x)⁢(Ω) and p+<∞, then the following properties hold:

We define the variable exponent Sobolev space by

On W1,p⁢(x)⁢(Ω) we may consider one of the following equivalent norms:

or

Let W01,p⁢(x)⁢(Ω) denote the closure of the set of compactly supported W1,p⁢(x)-functions with respect to the norm ∥u∥p⁢(x). When smooth functions are dense, we can also use the closure of C0∞⁢(Ω) in W1,p⁢(x)⁢(Ω). Using the Poincaré inequality, we can define the space W01,p⁢(x)⁢(Ω), in an equivalent manner, as the closure of C0∞⁢(Ω) with respect to the norm

The vector space (W01,p⁢(x)⁢(Ω),∥⋅∥) is a separable and reflexive Banach space. Moreover, if 0<|Ω|<∞ and p1, p2 are variable exponents so that p1≤p2 in Ω, then there exists the continuous embedding

Set

If (un),u∈W01,p⁢(x)⁢(Ω), then the following properties hold:

Set

We point out that if p,q∈C+⁢(Ω¯) and q⁢(x)<p⋆⁢(x) for all x∈Ω¯, then the embedding W01,p⁢(x)⁢(Ω)↪Lq⁢(x)⁢(Ω) is compact.

For a constant function p the variable exponent Lebesgue and Sobolev spaces coincide with the standard Lebesgue and Sobolev spaces. As pointed out in [15], the function spaces with variable exponent have some striking properties such as the following:

1. If 1<p-≤p+<∞ and p:Ω¯→[1,∞) is smooth, then the formula

   ∫ Ω | u ⁢ ( x ) | p ⁢ 𝑑 x = p ⁢ ∫ 0 ∞ t p - 1 ⁢ | { x ∈ Ω : | u ⁢ ( x ) | > t } | ⁢ 𝑑 t

   has no variable exponent analogue.

2. Variable exponent Lebesgue spaces do not have the mean continuity property. More precisely, if p is continuous and nonconstant in an open ball B, then there exists a function u∈Lp⁢(x)⁢(B) such that u⁢(x+h)∉Lp⁢(x)⁢(B) for all h∈ℝN with arbitrary small norm.

3. The function spaces with variable exponent are never translation invariant. The use of convolution is also limited, for instance the Young inequality

   | f * g | p ⁢ ( x ) ≤ C ⁢ | f | p ⁢ ( x ) ⁢ ∥ g ∥ L 1

   holds if and only if p is constant.

3 Spectrum Consisting in an Unbounded Interval

Assume p1,p2∈C+⁢(Ω¯) and consider the functions ϕ,ψ:Ω×[0,∞)→[0,∞) satisfying the following hypotheses:

1. The mappings ϕ⁢(⋅,ξ) and ψ⁢(⋅,ξ) are measurable on Ω for all ξ≥0 and ϕ⁢(x,⋅) and ψ⁢(x,⋅) are locally absolutely continuous on [0,∞) for almost all x∈Ω.

2. There exist functions a1∈Lp1′⁢(Ω) and a2∈Lp2′⁢(Ω) and b>0 such that

   | ϕ ⁢ ( x , | v | ) ⁢ v | ≤ a 1 ⁢ ( x ) + b ⁢ | v | p 1 ⁢ ( x ) - 1 , | ψ ⁢ ( x , | v | ) ⁢ v | ≤ a 2 ⁢ ( x ) + b ⁢ | v | p 2 ⁢ ( x ) - 1

   for almost all x∈Ω and for all v∈ℝN.

3. There exists c>0 such that

   ϕ ⁢ ( x , ξ ) ≥ c ⁢ ξ p 1 ⁢ ( x ) - 2 , ϕ ⁢ ( x , ξ ) + ξ ⁢ ∂ ⁡ ϕ ∂ ⁡ ξ ⁢ ( x , ξ ) ≥ c ⁢ ξ p 1 ⁢ ( x ) - 2

   and

   ψ ⁢ ( x , ξ ) ≥ c ⁢ ξ p 2 ⁢ ( x ) - 2 , ψ ⁢ ( x , ξ ) + ξ ⁢ ∂ ⁡ ψ ∂ ⁡ ξ ⁢ ( x , ξ ) ≥ c ⁢ ξ p 2 ⁢ ( x ) - 2

   for almost all x∈Ω and for all ξ>0.

Assume that q∈C+⁢(Ω¯) and

1. p1⁢(x)<q-≤q+<p2⁢(x)<p1*⁢(x) for all x∈Ω¯.

Let f:Ω×ℝ→ℝ be a Carathéodory function such that the following assumptions are fulfilled:

1. We have t⁢f⁢(x,t)≥0 for a.a. (x,t)∈Ω×ℝ and there exists m∈L∞⁢(Ω)+∖{0} such that

   | f ⁢ ( x , t ) | ≤ m ⁢ ( x ) ⁢ | t | q ⁢ ( x ) - 1   for a.a.  ⁢ x ∈ Ω ⁢  and all  ⁢ t ∈ ℝ .

2. There exist M>0 and θ>p1+ such that

   0 < θ ⁢ F ⁢ ( x , t ) ≤ t ⁢ f ⁢ ( x , t )   for a.a.  ⁢ x ∈ Ω ⁢  and all  ⁢ t ∈ ℝ ∖ { 0 } ,

   where F⁢(x,t):=∫0tf⁢(x,s)⁢𝑑s.

Consider the following nonlinear eigenvalue problem:

Problem (3.1) is driven by non-homogeneous operators of the type div⁡(ϕ⁢(x,|∇⁡u|)⁢∇⁡u). If ϕ⁢(x,ξ)=ξp⁢(x)-2, then we obtain the standard p⁢(x)-Laplace operator, that is, Δp⁢(x)⁢u:=div⁡(|∇⁡u|p⁢(x)-2⁢∇⁡u). Our abstract setting includes the case ϕ⁢(x,ξ)=(1+|ξ|2)(p⁢(x)-2)/2, which corresponds to the generalized mean curvature operator

The capillarity equation corresponds to

hence the corresponding capillary phenomenon is described by the differential operator

We say that u∈W01,p2⁢(x)⁢(Ω)∖{0} is a solution of problem (3.1) if

for all v∈W01,p2⁢(x)⁢(Ω).

In this case, u is an eigenfunction of problem (3.1) and the corresponding λ∈ℝ is an eigenvalue of (3.1). The choice of W01,p2⁢(x)⁢(Ω) as a suitable function space for problem (3.1) is dictated by our hypothesis (Q).

For ϕ and ψ described in hypotheses (H1)–(H3) we set

An important role in the proof of our main result is played by the following assumption, which is also used in [8] for the existence of weak solutions in a different framework:

1. For all x∈Ω¯ and all ξ∈ℝN the following estimate holds:

   0 ≤ [ ϕ ⁢ ( x , | ξ | ) + ψ ⁢ ( x , | ξ | ) ] ⁢ | ξ | 2 ≤ p 1 + ⁢ A 0 ⁢ ( x , | ξ | ) .

We notice that our hypothesis (f1) implies that

We define the following Rayleigh-type quotients:

and

We do not have any information about the contribution of real parameters satisfying λ∈[λ*,λ*) even in simple cases, for instance if Ω is a ball or for particular values of ϕ, ψ and f.

Related concentration properties are established in Kim and Kim [8], Mihăilescu and Rădulescu [11, 12], Rădulescu [14] and Rădulescu and Repovš [15, Chapter 3], see also Cencelj, Repovš, Virk [2] and Repovš [16] for recent contributions to anisotropic elliptic problems.

4 Proof of Theorem 3.1 and Perspectives

We first give the proof of our main result. For this purpose we establish several auxiliary results.

Define the functional A:W01,p2⁢(x)⁢(Ω)→ℝ by

where A0 is defined in (3.2).

Then A∈C1⁢(W01,p2⁢(x)⁢(Ω),ℝ) and for all u,v∈W01,p2⁢(x)⁢(Ω) we have

Moreover, the operator A:W01,p2⁢(x)⁢(Ω)→(W01,p2⁢(x)⁢(Ω))* is strictly monotone and is a mapping of type (S+), that is, if

and

then

Standard arguments also show that A is weakly lower semicontinuous. We refer to [8, Lemmas 3.2 and 3.4] for details and proofs.

Set

Then u∈W01,p2⁢(x)⁢(Ω)∖{0} is a solution of problem (3.1) if and only if A′⁢(u)=λ⁢B′⁢(u).

The next step is to show that the infimum in W01,p2⁢(x)⁢(Ω) of the Rayleigh quotient A⁢(u)/B⁢(u) is attained.

Recall that