Existence of normalized solutions for the Schrödinger equation

Abstract

In this paper, we devote to studying the existence of normalized solutions for the following Schrödinger equation with Sobolev critical nonlinearities. $ \begin{align*} &\left\{\begin{array}{ll} -\Delta u = \lambda u+\mu\lvert u \rvert^{q-2}u+\lvert u \rvert^{p-2}u&{\mbox{in}}\ \mathbb{R}^N,\\ \int_{\mathbb{R}^N}\lvert u\rvert^2dx = a^2, \end{array}\right. \end{align*} $ where $ N\geqslant 3 $, $ 2 < q < 2+\frac{4}{N} $, $ p = 2^* = \frac{2N}{N-2} $, $ a, \mu > 0 $ and $ \lambda\in\mathbb{R} $ is a Lagrange multiplier. Since the existence result for $ 2+\frac{4}{N} < p < 2^* $ has been proved, using an approximation method, that is let $ p\rightarrow 2^* $, we obtain that there exists a mountain-pass type solution for $ p = 2^* $.

Keywords

• normalized solutions
• schrödinger equation
• sobolev critical nonlinearities
• approximation method
• mountain-pass type solution