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Normalized solutions for p-Laplacian equation with general nonlinearities on bounded domain
Published online by Cambridge University Press: 13 November 2025
Abstract
We consider the existence, nonexistence and multiplicity of normalized solutions to the 
$p$-Laplacian equation on a bounded domain(0.1)
\begin{equation}\left\{\begin{array}{ll}-\Delta_p u+\lambda |u|^{p-2}u=g(u),&\text{in }\Omega,\\\int_\Omega |u|^p=\rho, u=0&\text{on }\partial \Omega,\end{array}\right.\end{equation}
where 
$\Omega$ is a bounded domain, 
$p\geq 2$. Firstly, under suitable assumptions on 
$\rho$, if 
$g$ is at most mass-critical at infinity, we prove the existence of infinitely many solutions. Secondly, for 
$\rho$ large, if 
$g$ is mass-supercritical, we perform a blow-up analysis to show the nonexistence of finite Morse index solutions. At last, for 
$\rho$ suitably small, combining with the monotonicity argument, we obtain a multiplicity result. In particular, when 
$p=2$, we obtain the existence of infinitely many normalized solutions.
Keywords
MSC classification
Primary:
35A01: Existence problems
Information
- Type
- Research Article
- Information
- Copyright
- © The Author(s), 2025. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh.
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