Nodal solutions for Laplace equations with critical Sobolev and Hardy exponents on $R^N$

Yinbin Deng; Qi Gao; Dandan Zhang

doi:10.3934/dcds.2007.19.211

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2007, Volume 19, Issue 1: 211-233. Doi: 10.3934/dcds.2007.19.211

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Nodal solutions for Laplace equations with critical Sobolev and Hardy exponents on $R^N$

1.
Department of Mathematics, Huazhong Normal University, Wuhan 430079
2.
Department of Mathematics, Xiangfan University, Xiangfan, 441053

Received: August 31, 2006

Revised: March 31, 2007

Published: December 2006

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Abstract

This paper is concerned with the existence and nodal character of the nontrivial solutions for the following equations involving critical Sobolev and Hardy exponents:

$-\Delta u + u - \mu \frac{u}{|x|^2}=|u|^{2^*-2}u + f(u),$
$u \in H^1_r (\R ^N),(1)$

where $2^$*$=\frac{2N}{N-2}$ is the critical Sobolev exponent for the embedding $H^1_r (\R ^N) \rightarrow L^{2^}$*$ (\R ^N)$, $\mu \in [0, \ (\frac {N-2}{2})^2)$ and $f: \R \rightarrow\R $ is a function satisfying some conditions. The main results obtained in this paper are that there exists a nontrivial solution of equation (1) provided $N\ge 4$ and $\mu \in [0, \ (\frac {N-2}{2})^2-1] $ and there exists at least a pair of nontrivial solutions $u^+_k$, $u^-_k$ of problem (1) for each k $\in N \cup \{0\}$ such that both $u^+_k$ and $u^-_k$ possess exactly k nodes provided $N\ge 6$ and $\mu \in [0, \ (\frac {N-2}{2})^2-4]$.

Keywords:

critical Sobolev and Hardy exponents; node solutions.

Mathematics Subject Classification: 35J25, 35J60, 35J65.

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Nodal solutions for Laplace equations with critical Sobolev and Hardy exponents on $R^N$

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