Positive solutions of a Schrödinger equation with critical nonlinearity (Q1881541)

The authors study the nonlinear Schrödinger equation transformed into the problem \[ -\triangle u+\lambda a(x)u=\mu u+u^{2^{\ast }-1}, \quad u\in \mathbb R^N, \;\lambda >0, \;\mu \in \mathbb R \] with critical exponent \(2^{\ast }=2N/(N-2)\), \(N\geq 4\), where \(a\geq 0\) has a potential well, and \(u>0\), \(u\in H^1(\mathbb R^N)\). Here \(a\) is a nonnegative and continuous function, and the Lebesgue measure in \(\mathbb R^N\) is defined by \(\mathcal L\)\(\{x\in \mathbb R^n:a(x)\leq M_0\}<\infty \). Having in mind the above stated conditions the authors prove existence and multiplicity of positive solutions which localize near the potential well for \(\mu \) small and \(\lambda \) large.