Solutions of a nonlinear Schrödinger equation
The authors of this interesting paper study the existence and multiplicity of solutions in \(H^1(\mathbb R^N)\) of nonlinear Schrödinger equations, \(\triangle u+V_{\lambda }(x)u=g(x,u)\) in \(\mathbb R^N\). It is assumed that \(u(x)\to 0\) as \(|x|\to \infty \), the potential \(V_{\lambda }(x)=\lambda a(x)+a_0(x)\), where \(\lambda >0\) is a real parameter, \(a_0\in L^{\infty }(\mathbb R^N)\), \(a\in L_{\text{loc}}^{\infty }(\mathbb R ^N)\), and there is a closed subset \(Z\subset \mathbb R^N\) with nonempty \(\Omega =\) int \(Z\) such that \(a(x)=0\) in \(Z\) and \(a(x)>0\) a.e. in \(\mathbb R ^N\setminus Z\), and there is \(M_0>0\) such that the Lebesgue measure mes\((\{x\in \mathbb R ^N:a(x)\leq M_0\})<\infty \). The authors improve some previous results in two respects. The first is that they do not require \(a_0\) to be positive. The second is that \(f(x,u)\) is a critical nonlinear function.
scientific article; zbMATH DE number 1751462
scientific article