Communications on Pure and Applied Analysis (CPAA)

Multiple solutions for a class of Ambrosetti-Prodi type problems for systems involving critical Sobolev exponents

Pages: 355 - 372, Volume 7, Issue 2, March 2008      doi:10.3934/cpaa.2008.7.355

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F. R. Pereira - Departamento de Matemática - ICE, Universidade Federal de Juiz de Fora, CEP: 36036-330, Juiz de Fora, Minas Gerais, Brazil (email)

Abstract: In this work we study the existence of multiple solutions for the non-homogeneous system

$ - \Delta U = AU + (u^p_+, v^p_+)+ F$ in $\Omega$

$ U = 0 $ on $ \partial\Omega,$

where $\Omega\subset \mathbb R^{N}$ is a bounded smooth domain; $U=(u,v), p=2^\star -1$, with $2^\star=\frac{2N}{N-2}, N \geq 3$; ${w_+}=$ max{ $w,0$} and $F \in L^s(\Omega)\times L^s(\Omega)$ for some $s>N$.
Using variational methods, we prove the existence of at least two solutions. The first is obtained explicitly by a direct calculation and the second via the Mountain Pass Theorem for the case $0< \mu_1 \leq \mu_2< \lambda_1$ or Linking Theorem if $\lambda_k < \mu_1 \leq \mu_2 < \lambda_{k+1}$, where $\mu_1, \mu_2$ are eigenvalues of symmetric matrix $A$ and $\lambda_j$ are eigenvalues of $(-\Delta, H_0^1(\Omega))$.

Keywords:  Ambrosetti-Prodi type problems, systems of elliptic equations, critical Sobolev exponents.
Mathematics Subject Classification:  35J50, 35B33.

Received: January 2007;      Revised: August 2007;      Available Online: December 2007.