Existence results for some nonlinear elliptic system with lack of compactness
The author investigates the existence of a weak solution of the system \[ -\Delta_p u=u\left| u\right|^{\alpha-1}\left| v\right|^{\beta+1}+f, \; -\Delta_p v=v\left| u\right|^{\alpha+1}\left| v\right|^{\beta-1}+g \] subject to homogeneous Dirichlet boundary conditions on a smooth bounded domain \(\Omega\subset {\mathbb R}^n\). It is supposed that \(\alpha,\;\beta>-1\), \({{(\alpha+1)(n-p)}\over{np}}+{{(\beta+1)(n- q)}\over{nq}}=1\), and \(\max\{p,q\}<\alpha+\beta+2\). The main result guarantees the existence of a solution, if \(f\in W^{-1,p'}(\Omega)\) and \(g\in W^{-1,q'}(\Omega)\) are both nonzero, and \(\left\| f\right\|_{-1,p'}+\left\| g\right\|_{-1,q'}\) is sufficiently small. The proof utilizes local minimization of an adapted problem. Convergence is obtained by concentration-compactness arguments. It should be noted that the nonlinearities involve critical Sobolev exponents.
scientific article; zbMATH DE number 1879994