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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References:

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