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

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  • 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 $(-\Delta, H_0^1(\Omega))$.

    Mathematics Subject Classification: 35J50, 35B33.


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