New type of solutions for the Nonlinear Schrödinger Equation in ${\mathbb{R}}^N$

Abstract

We construct a new family of entire solutions for the nonlinear Schrödinger equation \begin{align*} \begin{cases} -Δu+ V(y ) u = u^p, \quad u>0, \quad \text{in}~ \mathbb{R}^N, \\[2mm] u \in H^1(\mathbb{R}^N), \end{cases} \end{align*} where $p\in (1,\frac{N+2}{N-2})$ and $N\ge 3$, and $V(y)=V(|y|)$ is a positive bounded radial potential satisfying $$ V(|y|) = V_0 + \frac{a}{|y|^m} + O( \frac{1}{|y|^{m+σ}} ), \quad {\mbox {as}} \quad |y| \to \infty , $$ for some fixed constants $V_0,a,\sigma >0$, and $m>1$. Our solutions have strong analogies with the doubling construction of entire finite energy sign-changing solution for the Yamabe equation.