Multiplicity of positive solutions for singular elliptic problems

Aleksandra Orpel

Mathematische Nachrichten, 2021, vol. 294, issue 12, 2396-2412

Abstract: We consider the following nonlinear singular elliptic equation Δu(x)+f(x,u(x))−b(x)(u(x))−α∥∇u(x)∥β+g(x)x·∇u(x)=0inΩ,\begin{equation*}\hskip7pc \Delta u(x)+f(x,u(x))-b(x)(u(x))^{-\alpha }\Vert \nabla u(x)\Vert ^{\beta }+ g(x)x\cdot \nabla u(x)=0 \ \ \text{in} \ \ \Omega ,\hskip-7pc \end{equation*}where n>2$n>2$, Ω:={x∈Rn;∥x∥>R}$\Omega :=\lbrace x\in \mathbb {R}^{n};\,\Vert x\Vert >R \rbrace$. Our main purpose is to prove the existence of a large number of positive solutions with the asymptotic decay u(x)=O(∥x∥2−n)$u(x)=O\big (\Vert x\Vert ^{2-n}\big )$ as ∥x∥→∞$\Vert x\Vert \rightarrow \infty$. We also investigate the rate of decay of ∇u$\nabla u$. These results are based on the sub and supersolution method and cover both sublinear and superlinear cases of f.

Date: 2021
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