Normalized solutions for pseudo-relativistic Schrödinger equations

Abstract

In this paper, we consider the existence and multiplicity of normalized solutions to the following pseudo-relativistic Schrödinger equations $ \begin{equation*} \left\{ \begin{array}{lll} \sqrt{-\Delta+m^2}u +\lambda u = \vartheta |u|^{p-2}v +|u|^{2^\sharp-2}v, & x\in \mathbb{R}^N, \ u>0, \\ \ \int_{{\mathbb{R}^N}}|u|^2dx = a^2, \end{array} \right. \end{equation*} $ where $ N\geq2, $ $ a, \vartheta, m > 0, $ $ \lambda $ is a real Lagrange parameter, $ 2 < p < 2^\sharp = \frac{2N}{N-1} $ and $ 2^\sharp $ is the critical Sobolev exponent. The operator $ \sqrt{-\Delta+m^2} $ is the fractional relativistic Schrödinger operator. Under appropriate assumptions, with the aid of truncation technique, concentration-compactness principle and genus theory, we show the existence and the multiplicity of normalized solutions for the above problem.

Keywords

• pseudo-relativistic schrödinger operator
• normalized solutions
• sobolev critical exponent