Extremal solution and Liouville theorem for anisotropic elliptic equations

2021/01/04 by Li, Yuan
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2101.00970

Abstract

We study the quasilinear Dirichlet boundary problem \ \beginaligned -Quamp;=λe^u in Ω
uamp;=0 on ∂Ω,
\endaligned . where λ>0 is a parameter, Ω⊂ℝ^N with N≥2 be a bounded domain, and the operator Q, known as Finsler-Laplacian or anisotropic Laplacian, is defined by Qu:=∑_i=1^N\frac∂∂ x_i(F(∇ u)F_ξ_i(∇ u)). Here, F_ξ_i=\frac∂ F∂ξ_i and F: ℝ^N→[0,+∞) is a convex function of C²(ℝ^N∖\0\), that satisfies certain assumptions. We derive the existence of extremal solution and obtain that it's regular, if N≤9. We also concern the Hénon type anisotropic Liouville equation, namely, -Qu=(F⁰(x))^αe^u\quadin ℝ^N where α>-2, N≥2 and F⁰ is the support function of K:=\x∈ℝ^N:F(x)<1\ which is defined by F⁰(x):=sup_{ξ∈ K}⟨ x,ξ⟩. We obtain the Liouville theorem for stable solutions and the finite Morse index solutions for 2≤ N<10+4α and 3≤ N<10+4α^- respectively, where α^-=min\α,0\.

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