Multiplicity of solutions for nonhomogeneuous nonlinear elliptic equations with critical exponents

Norimichi Hirano

DOI: http://dx.doi.org/10.12775/TMNA.2001.034

Abstract


Let $N \geq 3$ and
$\Omega \subset \mathbb R^{N} $ be a bounded domain with a smooth
boundary $\partial \Omega $. We
%In this paper we
consider a semilinear boundary value problem of the form
%existence and multiplicity of solutions of problem
$$
\cases
-\Delta u = \vert u\vert ^{2^*-2} u +f &\text{\rm in } \Omega,\\
u> 0 & \text{\rm in } \Omega, \\
u= 0 & \text{\rm on } \partial \Omega,
\endcases
\leqno \text{\rm (P)}
$$
where $f \in C(\overline \Omega )$ and $2^* = 2N/(N-2)$. We show the effect of topology of
$\Omega $ on the multiple existence of solutions.

Keywords


Critical Sobolev; Dirichlet boundary value problem; semilinear elliptic problem

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