Topological Methods in Nonlinear Analysis

Existence of solutions for the semilinear corner degenerate elliptic equations

Jae-Myoung Kim

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Abstract

In this paper, we are concerned with the following elliptic equations: \begin{equation*}\label{e:JG} \begin{cases} -\Delta_{\mathbb{M}}u = \lambda f &\text{in } z:= (r,x,t) \in \mathbb{M}_0,\\ u= 0 &\text{on } \partial\mathbb{M}. \end{cases} \end{equation*} Here, $\lambda >0$ and $M=[0,1)\times X\times[0,1)$ as a local model of stretched corner-manifolds, that is, the manifolds with corner singularities with dimension $N=n+2\geq 3$. Here $X$ is a closed compact submanifold of dimension $n$ embedded in the unit sphere of $\mathbb{R}^{n+1}$. We study the existence of nontrivial weak solutions for the semilinear corner degenerate elliptic equations without the Ambrosetti and Rabinowitz condition via the mountain pass theorem and fountain theorem.

Article information

Source
Topol. Methods Nonlinear Anal., Volume 52, Number 2 (2018), 585-597.

Dates
First available in Project Euclid: 6 November 2018

Permanent link to this document
https://projecteuclid.org/euclid.tmna/1541473237

Digital Object Identifier
doi:10.12775/TMNA.2018.021

Mathematical Reviews number (MathSciNet)
MR3915652

Zentralblatt MATH identifier
07051681

Citation

Kim, Jae-Myoung. Existence of solutions for the semilinear corner degenerate elliptic equations. Topol. Methods Nonlinear Anal. 52 (2018), no. 2, 585--597. doi:10.12775/TMNA.2018.021. https://projecteuclid.org/euclid.tmna/1541473237


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