Concentration Phenomena of Normalized Solutions of Critical Biharmonic Equations with Combined Nonlinearities in $\mathbb{R}^{N}$

TL;DR

This paper investigates the existence, multiplicity, and concentration behavior of normalized solutions to critical biharmonic equations with combined nonlinearities in high-dimensional Euclidean space, employing variational methods and concentration-compactness techniques.

Contribution

It establishes the first results on concentration and multiplicity of normalized solutions for critical biharmonic equations with combined nonlinearities in ewline $ b{R}^N$, using minimization, truncation, and concentration-compactness methods.

Findings

Number of solutions is at least the number of global minima of the potential V.

Solutions concentrate around minima of V as the parameter ε approaches zero.

The study extends understanding of biharmonic equations with critical growth and combined nonlinearities.

Abstract

We prove the multiplicity and concentration of normalized solutions of critical biharmonic equations with combined nonlinearities in $\mathbb{R}^{N}$ \begin{equation*} \Delta^{2}u+V(\varepsilon x)u=\lambda u+\mu |u|^{q-2}u+|u|^{2^{**}-2}u \mbox{ in }\ \mathbb{R}^{N}, \quad \int_{\mathbb{R}^{N}}|u|^{2}dx=c^{2}, \end{equation*} where $\Delta^{2}$ is the biharmonic operator, $N\geq5$, $\mu,c>0$, $\varepsilon>0,$ $\lambda\in\mathbb{R}$, $q\in(2,2+\frac{8}{N}),$ and $2^{**}=\frac{2N}{N-4}$ is the Sobolev critical exponent. The potential $V$ is a bounded and continuous nonnegative function, satisfying some suitable global conditions. Using minimization techniques and a truncation argument, we show that the number of normalized solutions is not less than the number of global minimum points of $V$ when the parameter $\varepsilon$ is sufficiently small. To overcome the loss of compactness of the energy functional due to the critical growth, we apply the concentration-compactness principle. To the best of our knowledge, this study is the first contribution regarding the concentration and multiplicity properties of normalized solutions of critical biharmonic equations with combined nonlinearities in $\mathbb{R}^{N}$. To some extent, the main results included in this paper complement several recent contributions to the study of biharmonic equations with combined nonlinearities.