Non résonance près de la première valeur propre d'un système elliptique quasilinéaire de type potentiel

Let $\Omega$ be a bounded regular domain in $\Bbb R^n$. In this paper, we study the following problem: find $u\in\displaystyle\prod^m_{i=1}W_0^{1,p_i}(\Omega)$ such that:
$$
-\Delta_{p_i}u_i=\frac{\partial F}{\partial x_i}(x,u) +h_i(x)\text{ in }\Omega,\quad 1\le i\le m \tag"(S)"
$$
where $\Delta_{p_i}u_i=\operatorname{div}(|\nabla u_i|^{p-2}\nabla u_i)$, $1 < p_i < +\infty$ and $h_i\in W^{-1,p'_i}(\Omega)$.

We associate to (S) the eigenvalue problem:
$$
-\Delta_p v_i=\lambda \alpha_i|v_i|^{\alpha_i-2}v_i\prod_{j\ne i}|v_j|^{\alpha_j}\tag"(VP)"
$$
where $\alpha=(\alpha_1,\ldots,\alpha_m)$ satisfies $\alpha_i > 0$ and $\displaystyle\sum^m_{i=1}\frac{\alpha_i}{p_i}=1$.

We obtain nonresonance results for (S).

Roughly speaking if
$$
\lim\sup\frac{F(x,s)}{|s|^{\alpha}} < \lambda_1
$$
where $\lambda_1$ is the first eigenvalue of (VP), we prove the existence of a solution of (S).