First order spectrum for elliptic system and nonresonance problem (Q1817773)

The paper is concerned with the quasilinear elliptic system \[ \begin{cases} -\Delta u= au+ bv+ f(x,u,v,\nabla u,\nabla v)\quad &\text{in }\Omega,\\ -\Delta v= bu+ dv+ g(x,u,v,\nabla u,\nabla v)\quad &\text{in }\Omega,\\ u= v= 0\quad \text{on }\partial\Omega,\end{cases} \] where \(\Omega\) is a bounded domain in \(\mathbb{R}^N\) \((N\geq 1)\), \(a\), \(b\) and \(c\) are given real numbers, and the nonlinearities \(f, g:\Omega\times \mathbb{R}^2\times \mathbb{R}^{2N}\to \mathbb{R}^N\) are assumed to be Carathéodory functions with subcritical growth with respect to \(\xi= (\xi_1,\xi_2)\in \mathbb{R}^{2N}\). The system with the nonlinearities independent of \(\nabla u\) and \(\nabla v\) has been studied by several authors. To the system is extended the notion of the first-order spectrum, which is defined to be the set of couples \((\beta,\alpha)\in \mathbb{R}^n\times \mathbb{R}\) such that \[ \begin{cases} -\Delta u= au+ bv+ \alpha u+\beta\cdot\nabla u\quad &\text{in } \Omega,\\ -\Delta v= bu+ dv+ \alpha v+\beta\cdot\nabla v\quad &\text{in }\Omega,\\ u= v= 0\quad &\text{on }\partial\Omega,\end{cases} \] has a nontrivial solution \((u,v)\in (H^1_0(\Omega))^2\). Then, under appropriate assumptions on \(f\) and \(g\), which assure nonresonance with respect to the first-order spectrum, the existence of solutions \((u,v)\in (H^1_0(\Omega))^2\) of the system is established.