Existence of positive solutions for a semilinear Schrödinger equation in R N $\mathbb{R}^{N}$

Abstract

In this paper, we study the existence of multi-bump solutions for the semilinear Schrodinger equation \(-\Delta u+(1+\lambda a(x))u=(1-\lambda b(x))|u|^{p-2}u\), \(\forall u\in H^{1}(\mathbb{R}^{N})\), where \(N\geq1\), \(2 2\) if \(N=2\) or \(N=1\), \(a(x)\in C(\mathbb{R}^{N})\) and \(a(x)>0\), \(b(x)\in C(\mathbb{R}^{N})\) and \(b(x)>0\). For any \(n\in\mathbb{N}\), we prove that there exists \(\lambda(n)>0\) such that, for \(0<\lambda<\lambda(n)\), the equation has an n-bump positive solution. Moreover, the equation has more and more multi-bump positive solutions as \(\lambda\rightarrow0\).