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Semilinear elliptic equations involving critical Sobolev exponents. (English) Zbl 0674.35033

The author considers the semilinear elliptic boundary value problem with variable coefficients: \(Eu=bu^ p+\lambda hu\) in \(\Omega\), \(u>0\) in \(\Omega\) and \(u=0\) on \(\partial \Omega\), where \(Eu\equiv -\partial_ i(a_{ij} \partial_ ju)\) is a symmetric uniformly elliptic operator, b and h are nonnegative nontrivial bounded functions. Furthermore, \(\Omega\) is a bounded domain in \({\mathbb{R}}^ n\), \(n\geq 3\) and \(p=(n+2)/(n-2)\) is the critical exponent. Under certain conditions of \(a_{ij}\), b and h, the author obtains some existence and nonexistence results. The proof is standard. However, some interesting examples are included.
Reviewer: C.F.Wang

MSC:

35J65 Nonlinear boundary value problems for linear elliptic equations
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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