Quasilinear elliptic equations with critical growth via perturbation method (Q1759794)

The authors use a perturbation method to prove existence of ground state solutions for equations of the modified nonlinear Schrödinger type in \(\mathbb{R}^N\). These are equations \[ \sum_{i,j=1}^N D_j \left( a_{ij}(u)D_i u\right) - \frac{1}{2} \sum_{i,j=1}^N D_s a_{ij}(u)D_i u D_j u - V(x) u +f(u) =0, \] where \(D_i = \frac{\partial}{\partial x_i}\) and \(D_s a_{ij} (s) = \frac{\partial}{\partial s} a_{ij}(s)\). The potential \(V\) is assumed positive and so that \(V(x) \leq V_\infty := \lim_{|x|\to \infty} V(x)\), and the nonlinear term is \(f(s) = |s|^{\frac{4N}{N-2}-2}s +|s|^{p-2}s\). The exponent \(\frac{4N}{N-2}\) is the critical exponent for these problems and \(p\) is subcritical. Existence of solutions is proved first for a perturbed problem with additional terms, and with \(V (x) \equiv V_\infty\) constant. Then the contribution of the additional terms is diminished and the existence of solutions is preserved for the potential well \(V\).