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Some existence results for superlinear elliptic boundary value problems involving critical exponents. (English) Zbl 0614.35035

This paper deals with the superlinear elliptic boundary value problem

$-{\Delta }u-\lambda u={u|u|}^{{2}^{*}-2}{\phantom{\rule{1.em}{0ex}}in\phantom{\rule{1.em}{0ex}}{\Omega };\phantom{\rule{1.em}{0ex}}u|}_{\partial {\Omega }}=0,$

where ${\Omega }$ is a smoothly bounded domain in ${R}^{n}$, $n>2$, $\lambda \in R$, ${2}^{*}=2n/\left(n-2\right)$ and ${2}^{*}$ is the limiting Sobolev exponent for the embedding ${H}_{0}^{1}\left({\Omega }\right)\to {L}^{p}\left({\Omega }\right)$. Solving this problem is equivalent to finding critical points in ${H}_{0}^{1}\left({\Omega }\right)$ of the energy functional

${I}_{\lambda }\left(u\right)=\left(1/2\right){\int }_{{\Omega }}{\left(|\nabla u|}^{2}-{\lambda |u|}^{2}\right)dx-\left(1/2\right){\int }_{{\Omega }}{|u|}^{{2}^{*}}dx·$

First, based on the global compactness theorem for the problem (1), the compactness question is discussed. Then some results about the existence and multiplicity of solutions to the above problem are obtained.

Reviewer: Xu Zhenyuan

MSC:
 35J65 Nonlinear boundary value problems for linear elliptic equations 35J20 Second order elliptic equations, variational methods 35D05 Existence of generalized solutions of PDE (MSC2000) 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems