Infinitely many high energy solutions for fourth-order elliptic equations with p-Laplacian in bounded domain
Authors
Y. Chahma
- School of Mathematics and Statistics , Central South University, Changsha, Hunan, 410083, PR China.
- Faculty of Mathematics, University of Science and Technology Houari Boumediene, PB 32, El-Alia, Bab Ezzouar, Algiers, 16111, Algeria.
H. Chen
- School of Mathematics and Statistics , Central South University, Changsha, Hunan, 410083, PR China.
Abstract
In this paper, we study the following fourth-order elliptic equation with p-Laplacian, steep potential well and sublinear perturbation:
\[\begin{cases}
\Delta^2u-\kappa\Delta_p u+\mu V(x)u=f(x, u)+ \xi(x) \vert u\vert^{q-2}u, \quad
&x\in \Omega,\\
u= \Delta u=0, \quad &\text{on }\partial\Omega,
\end{cases}\]
where \(N \geq 5\), \(\Omega\) is a bounded domain in \(\mathbb{R}^N\) with smooth boundary \(\partial\Omega\), \(\Delta^2:=\Delta(\Delta)\) is the biharmonic operator, \(\Delta_p u= \operatorname{div}\left(\vert \nabla u\vert^{p-2} \nabla u\right)\) with \(p>2\), \(\mu,\kappa>0\) are parameters, \(f \in \mathcal{C}\left(\Omega \times \mathbb{R}, \mathbb{R}\right)\), \(\xi\in L^{\frac{2}{2-q}}\left(\Omega\right)\) with \(1\leq q<2\), we have the potential \(V\in \mathcal{C}(\Omega,\mathbb{R})\). Using variational methods, we establish the existence of infinitely many nontrivial high energy solutions under certain assumptions on \(V\) and \(f\).
Share and Cite
ISRP Style
Y. Chahma, H. Chen, Infinitely many high energy solutions for fourth-order elliptic equations with p-Laplacian in bounded domain, Journal of Mathematics and Computer Science, 32 (2024), no. 2, 109--121
AMA Style
Chahma Y., Chen H., Infinitely many high energy solutions for fourth-order elliptic equations with p-Laplacian in bounded domain. J Math Comput SCI-JM. (2024); 32(2):109--121
Chicago/Turabian Style
Chahma, Y., Chen, H.. "Infinitely many high energy solutions for fourth-order elliptic equations with p-Laplacian in bounded domain." Journal of Mathematics and Computer Science, 32, no. 2 (2024): 109--121
Keywords
- Variational methods
- p-Laplacian
- fourth-order elliptic equations
MSC
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