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Global attractors for a three-dimensional conserved phase-field system with memory
Multiple solutions for a class of Ambrosetti-Prodi type problems for systems involving critical Sobolev exponents
| 1. | Departamento de Matemática - ICE, Universidade Federal de Juiz de Fora, CEP: 36036-330, Juiz de Fora, Minas Gerais, Brazil |
$ - \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 $(-\Delta, H_0^1(\Omega))$.
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