Multiplicity results for a superlinear elliptic system with partial interference with the spectrum (Q880299)

In the present study the author deals with the following system \[ -\Delta_x u= au+ bv+ (v_+)^p+ f_1+ t\varphi_1\quad\text{in }\Omega, \] \[ -\Delta_x v= cu+ dv+ (u_+)^q+ f_2+ r\varphi_1\quad\text{in }\Omega, \] \[ u= v= 0\quad\text{on }\partial\Omega, \] where \(u_+(x)= \max\{0,u(x)\}\), \(\varphi_1> 0\) is the first eigenfunction of the Laplacian with Dirichlet boundary condition, and \(\Omega\subset\mathbb{R}^N\) is a smooth bounded domain, \(N\leq 2\). The nonlinearities are assumed both superlinear and subcritical, that is, \(1< p\), \(q< 2^*-1\), where \(2^*= {2N\over N-2}\) if \(N\geq 3\) and \(2^*= \infty\) if \(N= 2\). Using variational method the author proves existence of two solutions.