Existence and multiplicity of nontrivial solutions for some biharmonic equations with p-Laplacian

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• 作者：Juntao Sun^a ; ^b ; ^{sunjuntao2008@163.com" class="auth_mail" title="E-mail the corresponding author} ; Jifeng Chu^c ; ^{jifengchu@126.com" class="auth_mail" title="E-mail the corresponding author} ; Tsung-fang Wu^d ; ^{tfwu@nuk.edu.tw" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Biharmonic equations ; p-Laplacian ; Variational methods ; Gagliardo&ndash ; Nirenberg inequality
• 刊名：Journal of Differential Equations
• 出版年：2017
• 出版时间：15 January 2017
• 年：2017
• 卷：262
• 期：2
• 页码：945-977
• 全文大小：416 K

文摘

We investigate a class of nonlinear biharmonic equations with <em>pem>-Laplacian where N≥1$N\ge 1$, β∈R$\beta \in \mathbb{R}$, λ>0$\lambda >0$ is a parameter and Δ_pu=div(|∇u|^p−2∇u)${\mathrm{\Delta }}_{p}u=\text{div}(|\mathrm{\nabla }u{|}^{p-2}\mathrm{\nabla }u)$ with p≥2$p\ge 2$. Unlike most other papers on this problem, we replace Laplacian with <em>pem>-Laplacian and allow <em>β  em> to be negative. Under some suitable assumptions on V(x)$V(x)$ and f(x,u)$f(x,u)$, we obtain the existence and multiplicity of nontrivial solutions for <em>λem> large enough. The proof is based on variational methods as well as Gagliardo–Nirenberg inequality.