Pohozaev identity for Finsler anisotropic problems

In this paper we derive the Pohozaev identity for quasilinear equations -div(B′(H(∇u))∇H(∇u))=g(x,u)inΩ,(E)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} -{\text {div}}(B'(H(\nabla u))\nabla H(\nabla u))=g(x, u) \quad \text{ in }\,\, \Omega , \quad \quad {(E)} \end{aligned}$$\end{document}involving the anisotropic Finsler operator -div(B′(H(∇u))∇H(∇u))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-{\text {div}}(B'(H(\nabla u))\nabla H(\nabla u))$$\end{document}. In particular, by means of fine regularity results on the vectorial field B′(H(∇u))∇H(∇u)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B'(H(\nabla u))\nabla H(\nabla u)$$\end{document}, we prove the identity for weak solutions and in a direct way.


Introduction and main results
In this paper we derive the Pohozaev identity for the Finsler anisotropic operator, defined for suitable smooth functions as div(B (H(∇u))∇H(∇u)), (1.1) where the functions B, H are defined below. As well known, in 1965 Pohozaev in his seminal paper [13] considered the following Dirichlet problem −Δu = g (u) in Ω u = 0 on Ω, (1.2) with Ω a bounded smooth domain of R N and g a continuous function on R. He proved that, for classical solutions u ∈ C 2 (Ω) ∩ C 1 (Ω) to (1.2), the following identity holds where G = g with G(0) = 0 and η is the outward unit normal vector to the boundary ∂Ω. Important extensions and further developments can be found in the literature. We recall here the first result in the quasilinear setting [14] where the authors obtained a corresponding identity for smooth C 2 extremals of general variational problems, including results for systems and higher order operators. Since in general, e.g. for the p-Laplacian, solutions are not of class C 2 , we may consider very important the improvement of the results in [14] obtained in [8], where the authors proved the Pohozaev identity for C 1 solutions of general quasilinear Dirichlet problems, by mean of a suitable and technical approximation argument. See also the contribution in [11]. Although to our purposes we could also try to follow the technique of [8], we have to consider the fact that the adaptation to the anisotropic case of such a technique is completely non trivial. On the contrary, by means of fine regularity results, we shall follow a more simple a direct proof. The main consequence of (1.3) is the nonexistence of nontrivial solution to (1.2) when g satisfies suitable assumptions and Ω is star-shaped with respect to the origin. This is quite a delicate issue that we will discuss for the anisotropic case in Sect. 2. The typical nonexistence result for the critical case is contained in Corollary 2.5.
From now on we consider the following quasilinear anisotropic elliptic equation where Ω ⊆ R N is a smooth domain and g is a nonlinearity satisfying the following assumption (h g ) g : Ω × R → R is a C 1 function on the domain Ω × R. From the mathematical point of view, the anisotropy is responsible for a more richer geometric structure than the classical Euclidean case. On the other hand different applications come from several real phenomena where anisotropic media naturally arise.
In all the paper we suppose the following classical structural assumptions on the anisotropic operator. The anisotropic function H in (1.1) is a Finsler norm and we assume that the functions B and H satisfy for any t > 0; and Since ∂B H 1 is compact, the uniform ellipticity of H is equivalent to ask (1.5). For details we refer to [3,4].
Under our assumptions the natural function space for solutions to equation (E) is W 1,p (Ω)∩L ∞ (Ω). Indeed the anisotropic operator − div(B (H(∇u)) ∇H(∇u)), when κ = 0 in (h B ), becomes degenerate (p > 2) or singular (1 < p < 2) in the critical set and solutions are not classical in general. We remark at this stage that, as a particular special case, the anisotropic operator contains the well known p-laplace operator: as well known [9,15], already in this case the optimal regularity of solutions to quasilinear equations with the −Δ p operator is C 1,α . In [4] the authors show that it is possible the apply the results in [9,15] to show that weak solutions to (E) are C 1,α (Ω) ∩ C 2 (Ω\Z) for some 0 < α < 1.
We notice that, in general, the optimal regularity for solutions to this type of problems is known to be C 1,α , see the recent paper [1] about the interior regularity for weak solutions to anisotropic quasilinear equations. Regarding the regularity of the second derivatives we mention that first results have been obtained in [2]. In particular it has been shown in [2] that the stress field B (H(∇u))∇H(∇u) ∈ W 1,2 loc (Ω) among some very stronger regularity estimates that lead to the optimal summability of the second derivatives. In the case B(t) = t p /p the fact that H p−1 (∇u)∇H(∇u) ∈ W 1,2 loc (Ω) has been also proved recently in [1] under suitable more general assumptions on the source term.
Taking into account these facts, the goal of this paper is twofold: on one hand is to get a Pohozaev identity for quasilinear problems involving the general anisotropic operator (1.1); on the other hand is to get a general Pohozaev identity for C 1 weak solutions to (E) and then avoid some possible restriction coming eventually from asking more regularity on the solution to (E). To the best of our knowledge this identity is new for quasilinear equations driven by a general Finsler anisotropic operator in the framework of weak solutions.
We mention [16] where the authors proved a Pohozaev identity for a quasilinear problem involving the anisotropic p-Laplace operator (B(t) = t p /p) and assuming C 2 regularity of the solutions.
We are ready now to say what we mean for C 1 weak solutions to (E). More specifically we give the following In the following we use η(x) = (η 1 , . . . , η n )(x) to denote the unit outward normal vector to ∂Ω at a point x ∈ ∂Ω. We will also use the notation Trough the paper we define the functions G and G xi on the domain Ω × R as Here below we state our main results Let Ω a bounded smooth domain and let u ∈ C 1 (Ω) be a weak solution to (E). Then we have We observe that if we take B(t) = t p /p and H(ξ) = |ξ|, p > 1, i.e. the case of the p-Laplacian operator and g(x, u) = g(u), we recover the well known classical Pohozaev identity (see [7,8,11,12]) given in the following equation Having proved Theorem 1.2 we can exploit it to deduce a result in the whole R N . We have the following The rest of the paper is devoted to the proof of our main results and other consequences.

Notation
Generic fixed and numerical constants will be denoted by C (with subscript in some case) and they will be allowed to vary within a single line or formula. The first step to show Theorem 1.2 is the recovering of a stronger formulation for weak solutions to (E). We do it in the next proposition. We also collect there some regularity results on the weak solution to (E) that are interesting in itself. We have the following Proof. We prove part (i) of the statement. By [4], a C 1 weak solution u to (E), is actually regular outside the critical set Z, see (1.6). That is u ∈ C 2 (Ω\Z). In what follows, we denote by ∇u xi and u xixj the second derivatives of u outside the set Z. We mean them extended to zero on Z. At the end of the proof we will see that these derivatives coincide with the distributional ones in Ω. For all n > 0, let us define J n : R + 0 → R by setting (2.13) and let h n : R + 0 → R defined as h n (t) = J n (t) t and h n (0) = 0. (2.14) Let us set ϕ n (x) = B (H(∇u))H ξj (∇u)h n (|∇u|).
with β ∈ [0, 1) and where D 2 u denotes the hessian of the solution u. The estimate in (2.16) is proved in [2], see in particular the bound provided by (12) of Proposition 3.3 in [2], with γ = 0. Actually the result in [2] is proved in a local version but it can be extended to all over any compact set of the domain via a standard finite covering argument. First of all recalling that H is 1-homogeneous, we have that ∇ ξ H is 0-homogeneous and therefore we obtain for all ξ ∈ R N \ {0}. For a similar argument, we get the next estimate for the hessian of H, i.e.
We will use such estimates in the following computations.
Hence, by (h B )-(iii) and the fact that H(ξ) is a norm equivalent to the euclidian one, i.e. there exist c 1 , c 2 > 0 such that we obtain that where we used that h n (t) ≤ 1 for all t ≥ 0, where χ A is the characteristic function of a measurable set A. For 1 < p < 2 we also have Therefore using (2.16), from (2.20) and (2.20) we deduce that A n,1 L 2 (E) ≤ C, with C not depending on n.
The second term A n,2 can be estimated as follows As for the case of A n,1 , we get that A n,2 L 2 (E) ≤ C. For the last term A n, 3 we have have the following Exploiting (2.14), by a straightforward computation and then we have |∇u|h n (|∇u|) ≤ 2. Therefore from (2.22) and therefore, as above, we deduce that A n,3 L 2 (E) ≤ C. Hence taking in to account the previous estimates from (2.15) we get with C not depending on n. Therefore ϕ n ϕ ∈ W 1,2 (E). Moreover because of the compact embedding in L 2 (E), up to a subsequence, ϕ n → ϕ a.e. in E. On the other hand ϕ n → B (H(∇u))H ξj (∇u) and hence Since ξ j , j = 1, . . . , N is arbitrary, we have that B (H(∇u))∇ ξ H(∇u) ∈ W 1,2 (E, R N ), namely (2.11).
Once we have (2.11) integrating by parts the left hand side of (1.7) we get (2.12).
The part (ii) of the statement follows exploiting arguments contained in [6]. So we skip it.

Remark 2.2.
Note that, if the nonlinearity g allows to exploit the Hopf Boundary Lemma (see [2,Theorem 4.5]), then the regularity results of Proposition 2.1 can be extended to the clousure of Ω. Moreover if g(x, s) > 0 the critical set Z has zero Lebesgue measure, that is |Z| = 0. In this case the second distributional u xixj coincide a.e. with the classical ones.
In the proof of Theorem 1.2 we shall use a similar approach used in the proof of a very fine version of the divergence theorem [5, Lemma A. 1.]. For the reader's convenience we state it.  ∂Ω), with δ small enought such that the unique nearest point property holds for Ω. Therefore let us consider a smooth subdomain D δ Ω in a such a way that ∂D δ ⊂ Ω δ . For simplicity of notation, we set D = D δ .
Using the multiplier (x · ∇u) in both side of this equation and integrating we obtain  (2.26) where in the last line we have used the divergence theorem. We point out that integration in the left hand side of (2.25), in particular the divergence theorem, can be applied thanks to the regularity result of the Proposition 2.1, in particular (2.11). Note that we are arguing in the interior of the domain so that the local Sobolev regularity of the field is sufficient to apply the divergence theorem. At the end we will use a limit argument to recover the original domain Ω. Here we use a similar approach of M. Cuesta and P. Takáč in the proof of Lemma 2.3, see [5]. Hence, taking also into account the Euler's theorem for homogeneous functions, we obtain (2.27) where in the last line we applied the divergence theorem one more time.
Note that the estimates in Proposition 2.1 actually show that all the vector field (B (H(∇u))∇H(∇u))(x · ∇u) is locally a W 1,2 field. The proof can be carried out repeating verbatim the one of Proposition 2.1 which is easier in this case and, therefore, we skip it.
The thesis follows now, as in [5], letting δ → 0 and using a continuity argument in each term of (2.28). Note that the domain D δ , and therefore the boundary ∂D δ , can be arbitrary chosen here, in such a way that the convergence of the integrals follows easy. In particular a simple way to deduce this is to move a little bit ∂D, towards the interior of the domain along the normal direction.
As well known, the domain it is called star-shaped with respect to the origin if for every x ∈ Ω the line segment 0x is contained in Ω. Moreover it is easy to see that x · η ≥ 0 on ∂Ω and that in particular where ∂Ω 1 is some subset of the boundary of positive measure. Finally we call a domain strictly star-shaped with respect to the origin, if x · η > 0 on the whole boundary ∂Ω. (2.31) Using the fact that u = 0 on ∂Ω and therefore ∇u(x) = u η (x)η there, we get where (see hypotheses (h H )) we used Euler's theorem and that for s ∈ R we have ∇H(sξ) = sign(s)∇H(ξ) (i.e. ∇H is absolutely 0-homogeneous function). Hence the right hand side of (2.31) reads as ∂Ω G(x, 0) − B(H(∇u)) + B (H(∇u))H(∇u) (x · η) ds.
Then, since the domain is star-shaped, by (2.29) the right hand side of (2.31) is indeed non-negative and therefore (2.30) holds true. By assumption, either u = 0 or ∇u = 0 in Ω. Therefore in any case, by Stampacchia's theorem (see for instance [10, Lemma 7.7]) we deduce that ∇u = 0 a.e. in Ω. Therefore, since u = 0 on ∂Ω we have that u ≡ 0 in Ω.
The hypothesis of Theorem 2.4, namely that (2.30) implies either u = 0 or ∇u = 0 may not always be applicable. This is the case of the critical problem (2.33) here below: indeed, for such a problem, the left hand side of (2.30) is identically zero. Neverthless as a further and important consequence of Theorem 1.2, we get the non existence of nontriavial solutions in bounded smooth star-shaped domain Ω, of the critical anisotropic p-Laplacian problem, i.e.