The existence of ground state solutions for critical H\'{e}non equations in $\mathbb{R}^N$

TL;DR

This paper proves the existence of ground state solutions for critical Hénon equations in rica, using Nehari manifold and Brezis-Nirenberg methods, for equations with various critical exponents.

Contribution

It establishes the existence of positive and nonnegative radial ground state solutions for critical He9non equations with multiple critical exponents, including the upper He9non-Sobolev critical exponent.

Findings

Existence of positive radial ground state solutions for single critical exponent.

Existence of nonnegative radial ground state solutions for multiple critical exponents.

Identification of the upper He9non-Sobolev critical exponent as the critical embedding threshold.

Abstract

In this paper we confirm that $2^*(\gamma)=\frac{2(N+\gamma)}{N-2}$ with $\gamma>0$ is exactly the critical exponent for the embedding from $H_r^1(\mathbb{R}^N)$ into $L^q(\mathbb{R}^N;|x|^\gamma)$($N\geqslant 3$) (see \cite{2007SWW-1,2007SWW-2}) and name it as the upper H\'enon-Sobolev critical exponent. Based on this fact we study the ground state solutions of critical H\'enon equations in $\mathbb{R}^N$ via the Nehari manifold methods and the great idea of Brezis-Nirenberg in \cite{1983BN}. We establish the existence of the positive radial ground state solutions for the problem with one single upper H\'enon-Sobolev critical exponent. We also deal with the existence of the nonnegative radial ground state solutions for the problems with multiple critical exponents, including Hardy-Sobolev critical exponents or Sobolev critical exponents or the upper H\'{e}non-Sobolev critical exponents.