An existence result for nonliner elliptic problems involving critical Sobolev exponent. (English) Zbl 0612.35053

The authors study the problem \[ -\Delta u-\lambda u=| u|^{2^*-2}u\quad on\quad \Omega;\quad u=0\quad on\quad \partial \Omega, \] where \(\Omega\) is a bounded domain in \({\mathbb{R}}^ n\), \(\lambda\) is a real parameter, \(2^*=2n/(n-2)\) is the critical Sobolev exponent for the embedding \(H^ 1_ 0(\Omega)\subset L_ p(\Omega)\). It is proved that in the case \(n\geq 4\), for any \(\lambda\geq 0\) there exists at least one nontrivial solution \(u\in H^ 1_ 0(\Omega)\).
Reviewer: M.Kučera


35J65 Nonlinear boundary value problems for linear elliptic equations
35J25 Boundary value problems for second-order elliptic equations
35J20 Variational methods for second-order elliptic equations
35D05 Existence of generalized solutions of PDE (MSC2000)
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