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

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.