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July  2009, 8(4): 1269-1289. doi: 10.3934/cpaa.2009.8.1269

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, Chile

Received  April 2008 Revised  January 2009 Published  March 2009

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.
Keywords: solution with multiple double-spikes, reduction method, Nonhomogeneous elliptic equation, critical Sobolev exponent, small holes..
Mathematics Subject Classification: Primary: 35B40, 35B45; Secondary: 35J60, 47J3.
Citation: Salomón Alarcón. Multiple solutions for a critical nonhomogeneous elliptic problem in domains with small holes. Communications on Pure and Applied Analysis, 2009, 8 (4) : 1269-1289. doi: 10.3934/cpaa.2009.8.1269
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