The existence of ground state solutions for critical Hénon equations in ${\mathbb{R}}^N$

Abstract

In this paper we confirm that $2^{\ast }(\gamma )=\frac{2(N+\gamma )}{N-2}$ with $\gamma >0$ is exactly the critical exponent for the embedding from $H_r^1({\mathbb{R}}^N)$ into $L^q({\mathbb{R}}^N;|x{|}^{\gamma })$($N⩾3$) (see \cite{2007SWW-1,2007SWW-2}) and name it as the upper Hénon-Sobolev critical exponent. Based on this fact we study the ground state solutions of critical Hénon 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 Hénon-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 Hénon-Sobolev critical exponents.