Multiplicity of solutions for nonhomogeneuous nonlinear elliptic equations with critical exponents

Abstract

Let $N \geq 3$ and $\Omega \subset \mathbb R^{N} $ be a bounded domain with a smooth boundary $\partial \Omega $. We consider a semilinear boundary value problem of the form $$ \begin{cases} -\Delta u = \vert u\vert ^{2^*-2} u +f &\text{\rm in } \Omega,\\ u> 0 & \text{\rm in } \Omega, \\ u= 0 & \text{\rm on } \partial \Omega, \end{cases} \leqno \text{\rm (P)} $$ 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.