Volume 4, Issue 1
Nontrivial Solutions for Some Semilinear Elliptic Equations with Critical Sobolev Exponents

J. Part. Diff. Eq.,4(1991),pp.77-96

Published online: 1991-04

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• Abstract
Let Ω be a bounded domain in R^4(n ≥ 4) with smooth boundary ∂Ω. We discuss the existence of nontrivial solutions of the Dirichlet problem {- Δu = a(x) |u|^{4/(a-2)}u + λu + g(x, u), \quad x ∈ Ω u = 0, \quad x ∈ ∂Ω where a(x) is a smooth function which is nonnegative on \overline{Ω} and positive somewhere, λ> 0 and λ ∉ σ(-Δ). We weaken the conditions on a(x) that are generally assumed in other papers dealing with this problem.
• Keywords

Semilinear elliptic equation Sobolev exponent Critical value Critical point (P &#8226 S) condition