A singularity as a break point for the multiplicity of solutions to quasilinear elliptic problems

Advances in Nonlinear Analysis

article

Salvador López-Martínez¹ 

[1] Universidad de Granada

关键词: Nonlinear elliptic equations;    Singular gradient terms;    Multiplicity of solutions;    Uniqueness of solution;   

DOI  :  10.1515/anona-2020-0056

学科分类：社会科学、人文和艺术（综合）

来源: De Gruyter

【 摘 要 】

In this paper we deal with the elliptic problem −Δu=λu+μ(x)|∇u|quα+f(x) in Ω,u>0 in Ω,u=0 on ∂Ω, $$\begin{array}{} \begin{cases} \displaystyle-{\it\Delta} u=\lambda u+\mu(x)\frac{|\nabla u|^q}{u^\alpha}+f(x)\quad &\text{ in }{\it\Omega}, \\ u \gt 0 \quad &\text{ in }{\it\Omega}, \\ u=0\quad &\text{ on }\partial{\it\Omega}, \end{cases} \end{array} $$ where Ω ⊂ ℝ N is a bounded smooth domain, 0 ≨ μ ∈ L ∞ ( Ω ), 0 ≨ f ∈ L p 0 ( Ω ) for some p 0 > N2 $\begin{array}{} \frac{N}{2} \end{array}$, 1 0 and α 0 and q – 1 < α ≤ 1. We thus complement the results in [1, 2], which are concerned with α = q – 1, and show that the value α = q – 1 plays the role of a break point for the multiplicity/uniqueness of solution.

【 授权许可】

CC BY