Normalized concentrating solutions to nonlinear elliptic problems

TL;DR

This paper establishes the existence of normalized solutions to nonlinear elliptic problems that concentrate at specific points as the mass parameter varies, with implications for Schrödinger equations and Mean Field Games.

Contribution

It introduces new existence results for normalized solutions concentrating at points, depending on the mass size and critical thresholds, in nonlinear elliptic problems.

Findings

Solutions concentrate at points as mass approaches zero or infinity.

Existence results depend on the relation between p and the dimension N.

Applicable to whole space and bounded domain cases.

Abstract

We prove the existence of solutions $(\lambda, v)\in \mathbb{R}\times H^{1}(\Omega)$ of the elliptic problem \[ \begin{cases} -\Delta v+(V(x)+\lambda) v =v^{p}\ &\text{ in $ \Omega, $} \ v>0,\qquad \int_\Omega v^2\,dx =\rho. \end{cases} \] Any $v$ solving such problem (for some $\lambda$) is called a normalized solution, where the normalization is settled in $L^2(\Omega)$. Here $\Omega$ is either the whole space $\mathbb R^N$ or a bounded smooth domain of $\mathbb R^N$, in which case we assume $V\equiv0$ and homogeneous Dirichlet or Neumann boundary conditions. Moreover, $1<p<\frac{N+2}{N-2}$ if $N\ge 3$ and $p>1$ if $N=1,2$. Normalized solutions appear in different contexts, such as the study of the Nonlinear Schr\"odinger equation, or that of quadratic ergodic Mean Field Games systems. We prove the existence of solutions concentrating at suitable points of $\Omega$ as the prescribed mass $\rho$ is either small (when $p<1+\frac 4N$) or large (when $p>1+\frac 4N$) or it approaches some critical threshold (when $p=1+\frac 4N$).