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

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.