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Semilinear elliptic degenerate equations with critical exponent

N. M. Tri, D. A. Tuan

Abstract

In this paper we are mainly concerned with nontrivial positive solutions to the Dirichlet problem for the degenerate elliptic equation \begin{gather} -\frac{\partial^2 u}{\partial x^2} -\left|x\right|^{2k}\frac{\partial^2 u}{\partial y^2}=|x|^{2k}u^p+f(x,y,u) \quad\text{ in }Ω, \ u=0 \quad\text{ on }\partialΩ,\label{equ0} \end{gather} where $Ω$ is a bounded domain with smooth boundary in $\mathbb{R}^2, Ω\cap \{x=0\}\ne \emptyset,$ $k\in\mathbb N,$ $f(x,y,0)=0,$ and $p=(4+5k)/k$ is the critical exponent. Recently, the equation (1) was investigated in [12] for the subcritical case based on a new result obtained in [17] on embedding theorem of weighted Sobolev spaces. In the critical case considered in this paper we will essentially use the optimal functions and constants found in [17]

Semilinear elliptic degenerate equations with critical exponent

Abstract

In this paper we are mainly concerned with nontrivial positive solutions to the Dirichlet problem for the degenerate elliptic equation \begin{gather} -\frac{\partial^2 u}{\partial x^2} -\left|x\right|^{2k}\frac{\partial^2 u}{\partial y^2}=|x|^{2k}u^p+f(x,y,u) \quad\text{ in }Ω, \ u=0 \quad\text{ on }\partialΩ,\label{equ0} \end{gather} where is a bounded domain with smooth boundary in and is the critical exponent. Recently, the equation (1) was investigated in [12] for the subcritical case based on a new result obtained in [17] on embedding theorem of weighted Sobolev spaces. In the critical case considered in this paper we will essentially use the optimal functions and constants found in [17]
Paper Structure (6 sections, 14 theorems, 123 equations)

This paper contains 6 sections, 14 theorems, 123 equations.

Table of Contents

  1. Introduction
  2. Main Results
  3. Embedding Proposition
  4. Proof of Theorem \ref{['Case1']}
  5. Proof of Theorem \ref{['Case2']}
  6. Remarks on nonexistence solution for Case 2

Key Result

Theorem 1

Let $\lambda\in\mathbb R, \beta>-1/2$ and

Theorems & Definitions (36)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Theorem 1
  • Theorem 2
  • Corollary 1
  • Proposition 1
  • proof
  • Remark 1
  • ...and 26 more