In this paper, we consider singular Finsler double phase problems with nonlinear boundary condition. The Finsler double phase operator is defined by
for with being the Musielak–Orlicz–Sobolev space and F being a positive homogeneous function such that and the Hessian matrix is positive definite for all . Furthermore, μ is a nonnegative bounded function and . If F coincides with the Euclidean norm, that is, for , then (1.1) reduces to the usual double phase operator given by
Also, if or , then (1.2) (similarly (1.1)) reduces to the (Finsler) p-Laplacian or the (Finsler) -Laplacian, respectively. The study of such operators and corresponding energy functionals are motivated by physical phenomena; see, for example, the work of Zhikov  (see also the monograph of Zhikov, Kozlov and Oleĭnik ) 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 , Byun and Oh , Colombo and Mingione [14, 15], De Filippis and Palatucci , Marcellini [42, 41], Ok [43, 44], Ragusa and Tachikawa  and the references therein.
On the other hand, the minimization of the functional 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 describes, for example, the specific polyhedral shape of crystal structures in solid crystals with sufficiently small grains, as shown by Dinghas  and Taylor . It is clear that 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  and Wang and Xia , both dealing with the Serrin-type overdetermined anisotropic problem, or to Farkas, Fodor and Kristály  who studied a sublinear Dirichlet problem of this type. Related works concerning anisotropic phenomena can be found in the works of Bellettini and Paolini , Belloni, Ferone and Kawohl , Della Pietra and Gavitone , Della Pietra, di Blasio and Gavitone , Della Pietra, Gavitone and Piscitelli , Farkas , Farkas, Kristály and Varga , Ferone and Kawohl  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 , , is a bounded domain with Lipschitz boundary , is the outer unit normal of Ω at the point , λ is a positive parameter and the following assumptions hold true:
and are Carathéodory functions and there exist , as well as nonnegative constants such that
where and are the critical exponents to p given by
The function is a positively homogeneous Minkowski norm with finite reversibility
Because we are looking for positive solutions and hypothesis (H2) concerns the positive semiaxis , without any loss of generality, we may assume that for all and for a.a. or , respectively. Moreover, note that we always have ; see for example Farkas, Kristály and Varga . It is clear that the Euclidean norm has finite reversibility. Finally, we observe that (1.5) implies that compactly, as shown in Section 2.
A function is called a weak solution of problem (1.4) if , for a.a. and if
is satisfied for all .
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.
Let hypotheses (H1)–(H3) be satisfied. Then there exists such that for every problem (1.4) has a nontrivial weak solution.
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 for . 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 and are not compact. We overcome these difficulties with a local analysis on a suitable closed convex subset of combined with a truncation argument.
We point out that and are not the critical exponents to the space . Indeed, from Fan  we know that is continuous while is the Sobolev conjugate function of ; see also Crespo-Blanco, Gasiński, Harjulehto and Winkert [16, Definition 2.18 and Proposition 2.18]. So far it is not known how explicitly looks like in the double phase setting. For the moment, and seem to be the best exponents (probably not optimal) and only continuous (in general noncompact) embeddings from and 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  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  for the corresponding Neumann problem. Furthermore, under a different treatment, Chen, Ge, Wen and Cao  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 . The current paper can be seen as a nontrivial extension of the one in  to the case of a nonlinear boundary condition including type of critical growth. In particular, we are able to cover the situation when and/or , which has not been considered in  where .
Also, for the p-Laplacian or the -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 -Laplacian problems, respectively. Existence results for singular Neumann–Laplace problems have been obtained by Lei  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 , El Manouni, Marino and Winkert , Fiscella , Fiscella and Pinamonti , Gasiński and Papageorgiou , Gasiński and Winkert [34, 35, 36], Liu and Dai , Papageorgiou, Rădulescu and Repovš , Perera and Squassina , 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 , Barletta and Tornatore , Faraci and Farkas , Papageorgiou, Rădulescu and Repovš , Papageorgiou and Winkert  and Zeng, Bai, Gasiński and Winkert .
In this section, we are going to mention the main facts about the Minkowski space and the properties about Musielak–Orlicz–Sobolev spaces.
To this end, let be a positively homogeneous Minkowski norm, that is, F is a positive homogeneous function such that and the Hessian matrix is positive definite for all . We point out that the pair 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 is given by
The pair is a quasi-metric space and in general it holds .
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 -type matrix and is a fixed vector such that . Note that
The pair is often called Randers space which describes the electromagnetic field of the physical space-time in general relativity; see Randers . 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].
Let be a positively homogeneous Minkowski norm. Then the following assertions hold true:
are strictly convex.
Furthermore, and stand for the usual Lebesgue spaces endowed with the norm for . The corresponding Sobolev spaces are denoted by and equipped with the norms
On the boundary of Ω, we consider the -dimensional Hausdorff (surface) measure σ and denote by the boundary Lebesgue space with norm . We know that the trace mapping
is compact for and continuous for , where 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 be the function defined by
where (1.5) is satisfied. Then the Musielak–Orlicz space is defined by
equipped with the Luxemburg norm
where the modular function is given by
From Colasuonno and Squassina [13, Proposition 2.14], we know that the space is a reflexive Banach space.
Furthermore, we define the seminormed space
which is endowed with the seminorm
Similarly, we define with the seminorm .
The Musielak–Orlicz–Sobolev space 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 (1.5) be satisfied and let and be the critical exponents to p; see (1.6). Then the following embeddings hold:
are continuous for all
is continuous for all
and compact for all
is continuous for all
and compact for all
Let be the nonlinear operator defined by
where is the duality pairing between and its dual space . 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.
The operator B defined by (2.1) is bounded, continuous and monotone (hence maximal monotone).
3 Proof of the Main Result
Let be the functional given by
Due to the presence of the singular term, it is easy to see that is not .
Throughout the paper, we denote by and the inverses of the Sobolev embedding constants of and , respectively. This means, in particular,
Moreover, we define the function given by
where is finite by (H3). Since Ψ is strictly decreasing, we know there exists a unique such that . In addition, for all .
We start with the study of the functional given by
The next proposition shows the sequentially weakly lower semicontinuity of the functional
on closed convex subsets of .
Let hypotheses (H1)–(H3) be satisfied. For every the restriction of I to the closed convex set , which is given by
is sequentially weakly lower semicontinuous.
Let and let be such that in . We are going to prove that
For we consider the truncation functions given by
Note that for all .
First, we observe that
The same argument leads to
Since is sequentially weakly lower semicontinuous and considering that
due to the weak convergence of in , for every we have
Applying the triangle inequality for the Minkowski norm F (see Bao, Chern and Shen [2, Theorem 1.2.2]) along with the convexity of the function , , we get the following inequality:
From (3.6), by taking and , respectively, we get
On the other hand, by the Brezis–Lieb lemma (see, e.g., Papageorgiou and Winkert [49, Lemma 4.1.22], we have
for all and for all .
First, we have
Applying the inequality
with and , we get
since if . Combining (3.9) and (3.10) leads to
because if . This proves the claim.
Thus, we may apply the Brezis–Lieb lemma along with the claim in order to obtain
since , and so .
The last convergence in (3.12) follows from Proposition 2.2 (ii) since and due to the boundedness of , as given in (1.5).
Hence, for κ large enough, taking (3.3)–(3.5), (3.7), (3.8), (3.11) and (3.12) into account, we have that
We observe that
By Lebesgue’s dominated convergence theorem, we get that
Finally, combining (3.13)–(3.15), we arrive at
By using this along with (3.1) and the fact that for all (see (3.2)), it follows that
which proves the assertion of the proposition. ∎
Taking into account assumption (H2) together with the compact embeddings for and for (see Proposition 2.2 (ii) and (iii)), it is quite standard to prove that the functional
is sequentially weakly lower semicontinuous on for every . This fact along with Proposition 3.1 leads to the following corollary.
Let hypotheses (H1)–(H3) be satisfied. For every and for every , the restriction of to the closed convex set is sequentially weakly lower semicontinuous.
Now we are going to prove Theorem 1.2. For this purpose, we introduce the functionals and given by
Proof of Theorem 1.2.
Let and let be as in Corollary 3.2. First, we define
There exist small enough such that
In order to prove (3.16), it is enough to find λ, such that
Taking for some , we easily see that
Therefore, if we pass to the limit as , then (3.17) holds if
Thus we have to verify (3.18) to get our claim. First, note that
The growth conditions in (H2), along with the continuous embeddings as well as , yield
Hence, we obtain
Now, we consider the function given by
We easily see that , and from L’Hospital’s rule we verify that . Moreover, since (see (H2)) and due to the continuity of Λ, we know that there exists small enough such that for all . Hence, we find such that
Let us set
If we now take and , then (3.18) is satisfied, and so (3.16). This proves the claim.
From the claim we know that there exists an element with such that
From Corollary 3.2 we know that is sequentially weakly lower semicontinuous. Therefore,
restricted to has a global minimizer with . Suppose that . Then we have from (3.19) that
which is a contradiction. We conclude that is a local minimizer for with for .
We claim that . Let be such that and let . Then we have
which implies for sufficiently small. Thus, .
Let us now prove that is nonnegative a.e. in Ω. First, we observe that and for sufficiently small. Using this fact, we have
From this, we conclude
However, from Proposition 2.1 (iii) we know that
This leads to
Therefore, , and so a.e. in Ω.
Let us now show that u is positive in Ω. We argue indirectly and suppose there is a set C with positive measure such that in C. Let with and let small enough such that and a.e. in Ω. We obtain
a contradiction. Hence, a.e. in Ω.
Next we want to show that
for all with .
Now, we choose with and fix a decreasing sequence such that . It is clear that the functions
are measurable and nonnegative. Moreover, we have
Applying Fatou’s lemma gives
Then, for large enough, we obtain
Passing to the limit as in the inequality above and using (3.22), we derive (3.20) and have
which shows (3.21). Note that it is sufficient to prove the integrability in (3.20) for nonnegative test functions .
Now, let be such that for all . Note that the function has a local minimum in zero. We apply again Proposition 2.1 (iii) in order to get
Finally, we need to show that u is a positive weak solution of (1.4). To this end, let and take the test function in (3.21). Taking (3.23) into account, we have