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 p-Laplacian where class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039616303266&_mathId=si2.gif&_user=111111111&_pii=S0022039616303266&_rdoc=1&_issn=00220396&md5=f046146b18f79cb805721277d205f9cf" title="Click to view the MathML source">N≥1class="mathContainer hidden">class="mathCode">$N\ge 1$, class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039616303266&_mathId=si175.gif&_user=111111111&_pii=S0022039616303266&_rdoc=1&_issn=00220396&md5=ae327ddfa9a672086c0bfffbf06313a6" title="Click to view the MathML source">β∈Rclass="mathContainer hidden">class="mathCode">$\beta \in \mathbb{R}$, class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039616303266&_mathId=si242.gif&_user=111111111&_pii=S0022039616303266&_rdoc=1&_issn=00220396&md5=6002b7d3b70cc9051c56c014ed38311f" title="Click to view the MathML source">λ>0class="mathContainer hidden">class="mathCode">$\lambda >0$ is a parameter and class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039616303266&_mathId=si11.gif&_user=111111111&_pii=S0022039616303266&_rdoc=1&_issn=00220396&md5=3fbb2fcca900127c244cf352886b0175" title="Click to view the MathML source">Δ_pu=div(|∇u|^p−2∇u)class="mathContainer hidden">class="mathCode">${\mathrm{\Delta }}_{p}u=\text{div}(|\mathrm{\nabla }u{|}^{p-2}\mathrm{\nabla }u)$ with class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039616303266&_mathId=si6.gif&_user=111111111&_pii=S0022039616303266&_rdoc=1&_issn=00220396&md5=70fd0c5438e947cee324b071a40d14e0" title="Click to view the MathML source">p≥2class="mathContainer hidden">class="mathCode">$p\ge 2$. Unlike most other papers on this problem, we replace Laplacian with p-Laplacian and allow β   to be negative. Under some suitable assumptions on class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039616303266&_mathId=si7.gif&_user=111111111&_pii=S0022039616303266&_rdoc=1&_issn=00220396&md5=7f50bc09cb42ee5a739c1aef0a8171d8" title="Click to view the MathML source">V(x)class="mathContainer hidden">class="mathCode">$V(x)$ and class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039616303266&_mathId=si8.gif&_user=111111111&_pii=S0022039616303266&_rdoc=1&_issn=00220396&md5=ee13f4fa526ec2718cb949ecd8aa6b0a" title="Click to view the MathML source">f(x,u)class="mathContainer hidden">class="mathCode">$f(x,u)$, we obtain the existence and multiplicity of nontrivial solutions for λ large enough. The proof is based on variational methods as well as Gagliardo–Nirenberg inequality.