TY - JOUR

T1 - Radial symmetry of solutions to anisotropic and weighted diffusion equations with discontinuous nonlinearities

AU - Dipierro, Serena

AU - Poggesi, Giorgio

AU - Valdinoci, Enrico

PY - 2022/4

Y1 - 2022/4

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85124814736&partnerID=8YFLogxK

UR - https://arxiv.org/pdf/2105.02424.pdf

U2 - 10.1007/s00526-021-02157-5

DO - 10.1007/s00526-021-02157-5

M3 - Article

AN - SCOPUS:85124814736

SN - 0944-2669

VL - 61

JO - Calculus of Variations and Partial Differential Equations

JF - Calculus of Variations and Partial Differential Equations

IS - 2

M1 - 72

ER -