A variational approach to noncooperative elliptic systems. (English) Zbl 0852.35039

The paper deals with the existence of non-trivial solutions for elliptic systems of the form \[ - \Delta u_1= \lambda u_1- \delta u_2+ f_1(x, u_1, u_2),\;- \Delta u_2= \delta u_1+ \gamma u_2- f_2(x, u_1, u_2)\text{ in }\Omega,\tag{1} \]
\[ u_{1|_{\partial \Omega}}= 0,\quad u_{2|_{\partial \Omega}}= 0,\tag{2} \] where \(\Omega\) is a bounded domain in \(\mathbb{R}^N\) and the numbers \(\lambda\), \(\gamma\), \(\delta\) are real parameters, \(\gamma\leq \lambda\), \(\delta> 0\). The functions \(f_1, f_2: \Omega\times \mathbb{R}^2\to \mathbb{R}\) are assumed to be Carathéodory functions, \(f_1(x, 0, 0)= 0\), \(f_2(x, 0, 0)= 0\), \(|f_1(x, u)|+ |f_2(x, u)|\leq c|u|^{p- 1}+ d\) \(\forall u= (u_1, u_2)\in \mathbb{R}^2\), a.e. \(x\in \Omega\), for some constants \(c, d> 0\) and \(2\leq p< 2N/(N- 2)\) if \(N\geq 3\) or \(2\leq p< +\infty\) if \(N= 1, 2\). Moreover, assume that there exists a function \({\mathcal F}: \Omega\times \mathbb{R}^2\to \mathbb{R}\) of class \(C^1\) in the variables \(u= (u_1, u_2)\) such that \((f_1, f_2)= \nabla {\mathcal F}\) with respect to \((u_1, u_2)\).


35J50 Variational methods for elliptic systems
35J65 Nonlinear boundary value problems for linear elliptic equations
Full Text: DOI


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