Singular Finsler Double Phase Problems with Nonlinear Boundary Condition

1 Introduction

In this paper, we consider singular Finsler double phase problems with nonlinear boundary condition. The Finsler double phase operator is defined by

for u∈W1,ℋ⁢(Ω) with W1,ℋ⁢(Ω) being the Musielak–Orlicz–Sobolev space and F being a positive homogeneous function such that F∈C∞⁢(ℝN∖{0}) and the Hessian matrix ∇2⁡(F2/2)⁡(x) is positive definite for all x≠0. Furthermore, μ is a nonnegative bounded function and 1<p<q<N. If F coincides with the Euclidean norm, that is, F⁢(ξ)=(∑i=1N|ξi|2)1/2 for ξ∈ℝN, then (1.1) reduces to the usual double phase operator given by

Also, if μ≡0 or infΩ¯⁡μ>0, then (1.2) (similarly (1.1)) reduces to the (Finsler) p-Laplacian or the (Finsler) (q,p)-Laplacian, respectively. The study of such operators and corresponding energy functionals are motivated by physical phenomena; see, for example, the work of Zhikov [59] (see also the monograph of Zhikov, Kozlov and Oleĭnik [37]) in order to describe models for strongly anisotropic materials. Related functionals to (1.2) have been studied intensively with respect to regularity properties of local minimizers; see the works of Baroni, Colombo and Mingione [4, 5, 6], Baroni, Kuusi and Mingione [7], Byun and Oh [10], Colombo and Mingione [14, 15], De Filippis and Palatucci [18], Marcellini [42, 41], Ok [43, 44], Ragusa and Tachikawa [52] and the references therein.

On the other hand, the minimization of the functional EF:H01⁢(Ω)→ℝ defined by

under certain constraints on perimeter or volume occurs in many subjects of mathematical physics. Here the minimizer corresponds to an optimal shape (or configuration) of anisotropic tension-surface. The minimization of the functional EF describes, for example, the specific polyhedral shape of crystal structures in solid crystals with sufficiently small grains, as shown by Dinghas [22] and Taylor [54]. It is clear that EF is the energy functional to the Finsler Laplacian given by

The Finsler Laplacian, given in (1.3), has been studied by several authors in the last decade. We refer, for example, to the papers of Cianchi and Salani [12] and Wang and Xia [55], both dealing with the Serrin-type overdetermined anisotropic problem, or to Farkas, Fodor and Kristály [27] who studied a sublinear Dirichlet problem of this type. Related works concerning anisotropic phenomena can be found in the works of Bellettini and Paolini [8], Belloni, Ferone and Kawohl [9], Della Pietra and Gavitone [20], Della Pietra, di Blasio and Gavitone [19], Della Pietra, Gavitone and Piscitelli [21], Farkas [26], Farkas, Kristály and Varga [28], Ferone and Kawohl [30] and the references therein.

In this paper, we combine the effect of a Finsler manifold and a double phase operator along with a singular term and a nonlinear boundary condition. More precisely, we study the following problem:

where Ω⊂ℝN, N≥2, is a bounded domain with Lipschitz boundary ∂⁡Ω, ν⁢(x) is the outer unit normal of Ω at the point x∈∂⁡Ω, λ is a positive parameter and the following assumptions hold true:

1. 0 < γ < 1 and

   (1.5) 1 < p < q < N , q < p * , 0 ≤ μ ⁢ ( ⋅ ) ∈ L ∞ ⁢ ( Ω ) .

2. g 1 : Ω × ℝ → ℝ and g2:∂⁡Ω×ℝ→ℝ are Carathéodory functions and there exist 1<θ1<p≤ν1<p*, p<ν2<p* as well as nonnegative constants a1,a2,b1 such that

   g 1 ⁢ ( x , s ) ≤ a 1 ⁢ s ν 1 - 1 + b 1 ⁢ s θ 1 - 1   for a.a.  ⁢ x ∈ Ω ⁢  and for all  ⁢ s ≥ 0 ,

   g 2 ⁢ ( x , s ) ≤ a 2 ⁢ s ν 2 - 1   for a.a.  ⁢ x ∈ ∂ ⁡ Ω ⁢  and for all  ⁢ s ≥ 0 ,

   where p* and p* are the critical exponents to p given by

   (1.6) p * := N ⁢ p N - p   and   p * := ( N - 1 ) ⁢ p N - p .

3. The function F:ℝN→[0,∞) is a positively homogeneous Minkowski norm with finite reversibility

   r F = max w ≠ 0 ⁡ F ⁢ ( - w ) F ⁢ ( w ) .

Because we are looking for positive solutions and hypothesis (H2) concerns the positive semiaxis ℝ+=[0,∞), without any loss of generality, we may assume that g1⁢(x,s)=g2⁢(x,s)=0 for all s≤0 and for a.a. x∈Ω or x∈∂⁡Ω, respectively. Moreover, note that we always have rF≥1; see for example Farkas, Kristály and Varga [28]. It is clear that the Euclidean norm has finite reversibility. Finally, we observe that (1.5) implies that W1,ℋ⁢(Ω)↪Lq⁢(Ω) compactly, as shown in Section 2.

From hypotheses (H1)–(H3), we know that the definition of a weak solution is well defined.

The main result in this paper is the following theorem.

To the best of our knowledge, this is the first work on a singular double phase problem with nonlinear boundary condition even in the Euclidean case, that is, when F⁢(ξ)=(∑i=1N|ξi|2)1/2 for ξ∈ℝN. The novelty of our paper is not only due to the combination of the Finsler double phase operator with a singular term and nonlinear boundary condition. Indeed, in (1.4) we also deal with a type of critical Sobolev nonlinearities, even on the boundary, related to the lower exponent p, as explained in (1.6). Such critical terms make the study of compactness of the energy functional related to (1.4) more intriguing, since the embeddings W1,ℋ⁢(Ω)↪Lp*⁢(Ω) and W1,ℋ⁢(Ω)↪Lp*⁢(∂⁡Ω) are not compact. We overcome these difficulties with a local analysis on a suitable closed convex subset of W1,ℋ⁢(Ω) combined with a truncation argument.

We point out that p* and p* are not the critical exponents to the space W1,ℋ⁢(Ω). Indeed, from Fan [24] we know that W1,ℋ⁢(Ω)↪Lℋ*⁢(Ω) is continuous while ℋ* is the Sobolev conjugate function of ℋ; see also Crespo-Blanco, Gasiński, Harjulehto and Winkert [16, Definition 2.18 and Proposition 2.18]. So far it is not known how ℋ* explicitly looks like in the double phase setting. For the moment, p* and p* seem to be the best exponents (probably not optimal) and only continuous (in general noncompact) embeddings from W1,ℋ⁢(Ω)↪Lp*⁢(Ω) and W1,ℋ⁢(Ω)↪Lp*⁢(∂⁡Ω) are available. So we call it “types of critical growth”.

For singular double phase problems with Dirichlet boundary condition there exists only a few works. Recently, Liu, Dai, Papageorgiou and Winkert [40] studied the singular problem

Based on the fibering method along with the Nehari manifold, the existence of at least two weak solutions with different energy sign is shown; see also [17] for the corresponding Neumann problem. Furthermore, under a different treatment, Chen, Ge, Wen and Cao [11] considered problems of type (1.7) and proved the existence of a weak solution having negative energy. Finally, the existence of at least one weak solution to the singular problem

has been shown by the first and the third author in [29]. The current paper can be seen as a nontrivial extension of the one in [29] to the case of a nonlinear boundary condition including type of critical growth. In particular, we are able to cover the situation when 1<p<2 and/or 1<q<2, which has not been considered in [29] where 2≤p<q.

Also, for the p-Laplacian or the (q,p)-Laplacian only a few works exist involving singular terms and Neumann/Robin boundary conditions. We refer to Papageorgiou, Rădulescu and Repovš [47, 48] for singular homogeneous Neumann p-Laplace problems and for singular Robin (q,p)-Laplacian problems, respectively. Existence results for singular Neumann–Laplace problems have been obtained by Lei [38] based on variational and perturbation methods.

Finally, the reader can find existence results for double phase problems without singular term in the papers of Colasuonno and Squassina [13], El Manouni, Marino and Winkert [23], Fiscella [31], Fiscella and Pinamonti [32], Gasiński and Papageorgiou [33], Gasiński and Winkert [34, 35, 36], Liu and Dai [39], Papageorgiou, Rădulescu and Repovš [46], Perera and Squassina [51], Zeng, Bai, Gasiński and Winkert [56, 58] and the references therein. For related works dealing with certain types of double phase problems, we refer to the works of Bahrouni, Rădulescu and Winkert [1], Barletta and Tornatore [3], Faraci and Farkas [25], Papageorgiou, Rădulescu and Repovš [45], Papageorgiou and Winkert [50] and Zeng, Bai, Gasiński and Winkert [57].

2 Preliminaries

In this section, we are going to mention the main facts about the Minkowski space (ℝN,F) and the properties about Musielak–Orlicz–Sobolev spaces.

To this end, let F:ℝN→[0,∞) be a positively homogeneous Minkowski norm, that is, F is a positive homogeneous function such that F∈C∞⁢(ℝN∖{0}) and the Hessian matrix ∇2⁡(F2/2)⁡(x) is positive definite for all x≠0. We point out that the pair (ℝN,F) is the simplest not necessarily reversible Finsler manifold whose flag curvature is identically zero, the geodesics are straight lines and the intrinsic distance between two points x,y∈ℝN is given by

The pair (ℝN,dF) is a quasi-metric space and in general it holds dF⁢(x,y)≠dF⁢(y,x).

The so-called Randers metric is a typical example for a Minkowski norm with finite reversibility, which is given by

where A is a positive definite and symmetric (N×N)-type matrix and b=(bi)∈ℝN is a fixed vector such that 〈A-1⁢b,b〉<1. Note that

The pair (ℝN,F) is often called Randers space which describes the electromagnetic field of the physical space-time in general relativity; see Randers [53]. They are deduced as the solution of the Zermelo navigation problem.

In the next proposition we recall some basic properties of F; see Bao, Chern and Shen [2, Section 1.2].

Furthermore, Lr⁢(Ω) and Lr⁢(Ω;ℝN) stand for the usual Lebesgue spaces endowed with the norm ∥⋅∥r for 1≤r<∞. The corresponding Sobolev spaces are denoted by W1,r⁢(Ω) and W01,r⁢(Ω) equipped with the norms

respectively.

On the boundary ∂⁡Ω of Ω, we consider the (N-1)-dimensional Hausdorff (surface) measure σ and denote by Lr⁢(∂⁡Ω) the boundary Lebesgue space with norm ∥⋅∥r,∂⁡Ω. We know that the trace mapping

is compact for r~<r* and continuous for r~=r*, where r* is the critical exponent of r on the boundary given by

For simplification, we will avoid the notation of the trace operator throughout the paper.

Let us now introduce the Musielak–Orlicz–Sobolev spaces. For this purpose, let ℋ:Ω×[0,∞)→[0,∞) be the function defined by

where (1.5) is satisfied. Then the Musielak–Orlicz space Lℋ⁢(Ω) is defined by

equipped with the Luxemburg norm

where the modular function ρℋ:Lℋ⁢(Ω)→ℝ is given by

From Colasuonno and Squassina [13, Proposition 2.14], we know that the space Lℋ⁢(Ω) is a reflexive Banach space.

Furthermore, we define the seminormed space

which is endowed with the seminorm

Similarly, we define Lμq⁢(Ω;ℝN) with the seminorm ∥F⁢(⋅)∥q,μ.

The Musielak–Orlicz–Sobolev space W1,ℋ⁢(Ω) is defined by

equipped with the norm

Finally, we mention the main embedding results between Musielak–Orlicz–Sobolev spaces and usual Lebesgue and Sobolev spaces. We refer to Gasiński and Winkert [36, Proposition 2.2] or Crespo-Blanco, Gasiński, Harjulehto and Winkert [16, Proposition 2.17].

Let B:W1,ℋ⁢(Ω)→W1,ℋ⁢(Ω)* be the nonlinear operator defined by

where 〈⋅,⋅〉ℋ,F is the duality pairing between W1,ℋ⁢(Ω) and its dual space W1,ℋ⁢(Ω)*. The operator

has the following properties (see Crespo-Blanco, Gasiński, Harjulehto and Winkert [16, Proposition 3.4 (ii)]) by taking the properties of F into account.

3 Proof of the Main Result

Let Jλ:W1,ℋ⁢(Ω)→ℝ be the functional given by

where u±=max⁡(±u,0) and

Due to the presence of the singular term, it is easy to see that Jλ is not C1.

Throughout the paper, we denote by cp* and cp* the inverses of the Sobolev embedding constants of W1,p⁢(Ω)↪Lp*⁢(Ω) and W1,p⁢(Ω)↪Lp*⁢(∂⁡Ω), respectively. This means, in particular,

Moreover, we define the function Ψ:(0,∞)→ℝ given by

where rF=maxw≠0⁡F⁢(-w)F⁢(w) is finite by (H3). Since Ψ is strictly decreasing, we know there exists a unique ϱ*>0 such that Ψ⁢(ϱ*)=0. In addition, Ψ⁢(s)≥0 for all s∈(0,ϱ*).

We start with the study of the functional I:W1,ℋ⁢(Ω)→ℝ given by

The next proposition shows the sequentially weakly lower semicontinuity of the functional

on closed convex subsets of W1,ℋ⁢(Ω).