On concentration phenomena of least energy solutions to nonlinear Schrödinger equations with totally degenerate potentials
This paper consider the NLS equation with an electric potential \(V\) \[ -h^{2}\Delta u+V\left( x\right) u=f\left( u\right) \text{ in }\mathbb{R}^{N},\text{ }u>0,\text{ }u\in H^{1}\left( \mathbb{R}^{N}\right),\tag{1} \] where \(h>0,\) the dimension \(N\geq1\), the potential \(V\) is assumed to satisfy the following: \(V\in C^{1}\left( \mathbb{R}^{N};\mathbb{R}\right) ,\) \(V\left( x\right) \rightarrow+\infty,\) as \(\left| x\right| \rightarrow+\infty,\) and \(\inf_{x\in\mathbb{R}^{N}}V\left( x\right) =1\) and \(f\) is the nonlinear term which in the simplest settings includes the case \(f\left( t\right) =t^{p}\) with \(1<p<\infty,\) if \(N=1,2,\) and \(1<p<\left( N+2\right) /\left( N-2\right) \) if \(N\geq3.\) The author studies the concentration phenomena of the least energy solution for the settled problem. More precisely, he studies the asymptotic location of the concentration point of the least energy solutions. The author extends the result of the paper [\textit{G. Lu} and \textit{J. Wei}, C. R. Acad. Sci., Paris, Sér. I, Math. 326, No. 6, 691--696 (1998; Zbl 0911.35047)] where the precise asymptotic location of the concentration point of the least energy solutions for (1) with \(f\left( t\right) =t^{p}\) were considered. This extension is in several directions. Firstly, the author considers more general nonlinearities \(f\left( u\right) .\) Also, he drops the connectivity condition of the interior of \(\Omega=\{\left. x\in \mathbb{R}^{N}\right| V\left( x\right) =\inf_{y\in\mathbb{R}^{N}}V\left( y\right) \}\) and he relaxes the assumptions on \(V\). The proof is based on a modification of the method developed in [\textit{M. del Pino} and \textit{P. L. Felmer}, Indiana Univ. Math. J. 48, No. 3, 883--898 (1999; Zbl 0932.35080)] employing the rearrangement technique, without using the so-called uniqueness-non degeneracy assumption.
scientific article; zbMATH DE number 6855948
scientific article