Multiple positive solutions of nonhomogeneous elliptic systems with strongly indefinite structure and critical Sobolev exponents. (Q1419772)

Let \(\Omega\) be a smooth bounded domain in \({\mathbb R}^N\) \((N\geq 3)\) and let \(\lambda\), \(\mu\) be positive real numbers. This paper establishes a multiplicity result for the semilinear elliptic system \(-\Delta v=\lambda u+u^p+\varepsilon f(x)\), \(-\Delta u=\mu v+v^q+\delta g(x)\) in \(\Omega\); \(u,v>0\) in \(\Omega\); \(u=v=0\) on \(\partial\Omega\), where \(f,g\in L^\infty (\Omega)\) are non-negative functions, while the exponents \(p\) and \(q\) satisfy \((p+1)^{-1}+(q+1)^{-1}=(N-2)/N\). Due to the lack of compactness and the strong indefiniteness of the quadratic part in the above equation, it is difficult to apply traditional direct variational methods to deduce the existence of solutions. That is why, using comparison techniques which rely on adaptations of the dual variational method, the authors prove the existence of at least two nontrivial positive solutions, provided that \(\lambda,\mu\in (0,\lambda_1)\) and \(\varepsilon,\delta\in (0,\delta_0)\), where \(\lambda_1\) denotes the first eigenvalue of the Laplace operator \((-\Delta)\) in \(H^1_0(\Omega)\), and \(\delta_0\) is a positive number.