We construct a new family of entire solutions for the nonlinear Schr\"odinger equation \begin{ align*} \begin{ cases} -\Delta u+ 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 $N\geq 3$, and $V (y)= V(|y|)$ is a positive bounded radial potential satisfying $$ V(|y|) = V_0 + \frac{ a}{ |y|^m} + O( \frac{ 1}{ |y|^{ m+\sigma}} ), \quad { \mbox { as}} \quad |y| \to \infty , $$ 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.
Categories: math.AP