Nonexistence and existence of solutions with prescribed norm for semilinear nonlocal elliptic equations

Abstract

In this paper, using Pohozaev's identity and discussing minimizing sequences for the related constrained minimization problem, we study the nonexistence and existence of solutions with prescribed $L^2$-norm for the following nonlocal equation: $-{({\int }_{{\mathbb{R}}^N}|u(x){|}^2\mathrm{d}x)}^{\gamma }\mathrm{\Delta }u-|u{|}^{p-2}u=\lambda u,x\text{ in }{\mathbb{R}}^N,$ where $N\ge 3$, $p\in [2,2^{\ast })$ and $2^{\ast }=\frac{2N}{N-2}$.

Keywords:

• Constrained minimization
• Pohozaev's identity
• nonlocal equation
• existence
• parabolic equations

MSC2010:

• 35K57
• 35B35

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work is supported by the National Natural Science Foundation of China (62073203) and the Fund of Natural Science of Shandong Province (ZR2018MA022).