On the Schrödinger equations with a nonlinearity in the critical growth (Q519372)

The paper considers a nonlinear Schrödinger equation of the form \[ -\Delta u + Vu = af(u) + u^{2^*-1}, \] where \(N\geq 3\) and \(2^* = \frac{2N}{N-2}\) is the Sobolev conjugate exponent. The goal is to establish the existence of ground state solutions under various assumptions on the coefficient functions \(V\) and \(a\), and the nonlinearity \(f\). The proof rests on variational techniques and some compactness property for the associated energy functional.