Multiple solutions for a critical nonhomogeneous elliptic problem in domains with small holes

Salomón Alarcón

doi:10.3934/cpaa.2009.8.1269

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2009, Volume 8, Issue 4: 1269-1289. Doi: 10.3934/cpaa.2009.8.1269

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Multiple solutions for a critical nonhomogeneous elliptic problem in domains with small holes

Salomón Alarcón^1,

1.
Departamento de Matemática, Universidad Técnica Federico Santa María, Casilla 110-V, Valparaíso

Received: March 31, 2008

Revised: December 31, 2008

Published: June 2009

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Abstract

We consider the problem: $-\Delta u=|u|^{\frac{4}{N-2}}u+\varepsilon f(x)$ in $\Omega$, $u=0$ on $\partial\Omega$, where $\Omega\subset R^N$ is a bounded smooth domain which exhibits small holes, $f\geq 0$, $f$ is not equivalent to $0$ and $\varepsilon>0$ is small. Using the reduction method and a min-max scheme worked out with topological arguments, we construct multiple solutions by gluing negative double-spike patterns located near each of the holes.

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Mathematics Subject Classification: Primary: 35B40, 35B45; Secondary: 35J60, 47J30.

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