Radial symmetry of solutions to anisotropic and weighted diffusion equations with discontinuous nonlinearities

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For 1 < p< ∞, we prove radial symmetry for bounded nonnegative solutions of {-div{w(x)H(∇u)p-1∇ξH(∇u)}=f(u)w(x)inΣ∩Ω,u=0onΓ0,⟨∇ξH(∇u),ν⟩=0onΓ1\{0},where Ω is a Wulff ball, Σ is a convex cone with vertex at the center of Ω , Γ : = Σ ∩ ∂Ω , Γ 1: = ∂Σ ∩ Ω , H is a norm, w is a given weight and f is a possibly discontinuous nonnegative nonlinearity. Given the anisotropic setting that we deal with, the term “radial” is understood in the Finsler framework, that is, the function u is radial if there exists a point x such that u is constant on the Wulff shapes centered at x. When Σ = RN, J. Serra obtained the symmetry result in the isotropic unweighted setting (i.e., when H(ξ) ≡ | ξ| and w≡ 1). In this case we provide the extension of his result to the anisotropic setting. This provides a generalization to the anisotropic setting of a celebrated result due to Gidas-Ni-Nirenberg and such a generalization is new even for p= 2 whenever N> 2. When Σ ⊊ RN the results presented are new even in the isotropic and unweighted setting (i.e., when H is the Euclidean norm and w≡ 1) whenever 2 ≠ p≠ N. Even for the previously known case of unweighted isotropic setting with p= 2 and Σ ⊊ RN, the present paper provides an approach to the problem by exploiting integral (in)equalities which is new for N> 2 : this complements the corresponding symmetry result obtained via the moving planes method by Berestycki-Pacella. The results obtained in the isotropic and weighted setting (i.e., with w≢ 1) are new for any p.

Original languageEnglish
Article number72
JournalCalculus of Variations and Partial Differential Equations
Issue number2
Publication statusPublished - Apr 2022


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