Positive solutions of a Schrödinger equation with critical nonlinearity. (English) Zbl 1060.35130

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.


35Q55 NLS equations (nonlinear Schrödinger equations)
35J65 Nonlinear boundary value problems for linear elliptic equations