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

Abstract

We prove the multiplicity and concentration of normalized solutions of critical biharmonic equations with combined nonlinearities in $\mathbb{R}^{N}$ \begin{equation*} Δ^{2}u+V(\varepsilon x)u=λu+μ|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 $Δ^{2}$ is the biharmonic operator, $N\geq5$, $μ,c>0$, $\varepsilon>0,$ $λ\in\mathbb{R}$, $q\in(2,2+\frac{8}{N}),$ 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.

Theorems & Definitions (43)

• Theorem 1.1
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Lemma 3.3
• Lemma 3.4