On a semilinear elliptic system (Q1891572)

The authors consider a Dirichlet problem for the elliptic system \[ - \Delta v= H_u(u, v),\;-\Delta u= H_v(u, v)\tag{\(*\)} \] in some bounded domain \(\Omega\subset \mathbb{R}^N\) with smooth boundary. The growth conditions on \(H\) are \(|H_u(u, v)|\leq C(1+ |u|^p+ |v|^{(q+ 1){p\over p+ 1}})\), \(|H_v(u, v)|\leq C(1+ |v|^q+ |u|^{(p+ 1){q\over q+ 1}})\) and \(1/(p+ 1)+ 1/(q+ 1)> (N- 2)/2\). The authors do not assume \(p, q> 1\) which leads them to consider the functional \({\mathcal L}(u, v):= \int_\Omega (\nabla u\nabla v- H(u, v))\) (of which \((*)\) are the Euler- Lagrange equations) in the Banach space \(E_r:= W^{1, r}_0\times W^{1, r'}_0\), \(1/r+ 1/r'= 1\) rather than in Hilbert spaces as in previous work. (Here \(r\) depends on \(p\) and \(q\).) Based on an abstract critical point theorem in a product \(X_1\times X_2\) of two real reflexive Banach spaces they establish existence of a nontrivial weak solution which is a strong one under appropriate regularity assumptions.