The Nehari manifold for indefinite semilinear elliptic systems involving critical exponent (Q387440)

The authors study the combined effect of concave and convex nonlinearities on the number of solutions for an indefinite semilinear elliptic system of the type NEWLINE\[NEWLINE\begin{cases} -\Delta u=f_\lambda(x)|u|^{q-2}u+{\alpha\over{\alpha+\beta}}h(x)|u|^{\alpha-2}u|v|^\beta &\text{in}\;\Omega,\\ -\Delta v=g_\mu(x)|v|^{q-2}v+{\beta\over{\alpha+\beta}}h(x)|u|^{\alpha}|v|^{\beta-2}v &\text{in}\;\Omega,\\ u=v=0 &\text{on}\;\partial\Omega, \end{cases} NEWLINE\]NEWLINE involving critical exponents and sign-changing weight functions. In particular, \(\Omega\subset\mathbb R^N\) is a bounded domain, \(N\geq3\), \(0\in\Omega\), \(\alpha\), \(\beta>1\), \(\alpha+\beta=2^\ast ={{2N}\over{N-2}}\), \(q\in(1,2),\) \(\lambda\), \(\mu\geq 0\). NEWLINENEWLINENEWLINEUsing the Nehari manifold, the authors prove that the system have at least two nontrivial nonnegative solutions when the pair of the parameters \((\lambda,\mu)\) belongs to a certain subset of \(\mathbb R^2\).