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dc.contributor.authorZhao, Ji-Hong ( )
dc.contributor.authorZhao, Peihao ( )
dc.identifier.citationZhao, J. H., & Zhao, P. H. (2007). Infinitely many weak solutions for a p-Laplacian equation with nonlinear boundary conditions. Electronic Journal of Differential Equations, 2007(90), pp. 1-14.en_US

We study the following quasilinear problem with nonlinear boundary conditions

pu + α(x)|u|p-2u = ƒ(x, u) in Ω,
|∇u|p-2 ∂u/∂v = g(x, u) on ∂Ω,

where Ω is a bounded domain in ℝN with smooth boundary and ∂/∂v is the outer normal derivative, Δpu = div(|∇u|p-2∇u) is the p-Laplacian with 1 < p < N. We consider the above problem under several conditions on ƒ and superlinear and subcritical with respect to u, then we prove the existence of infinitely many solutions of this problem by using "fountain theorem" and "dual fountain theorem" respectively. In the case, where g is superlinear but subcritical and ƒ is critical with a subcritical perturbation, namely ƒ(x, u) = |u|p*-2u + λ|u|r-2u, we show that there exists at least a nontrivial solution when p < r < p* and there exist infinitely many solutions when 1 < r < p, by using "mountain pass theorem" and concentration-compactness principle" respectively.

dc.format.extent14 pages
dc.format.medium1 file (.pdf)
dc.publisherTexas State University-San Marcos, Department of Mathematicsen_US
dc.sourceElectronic Journal of Differential Equations, 2007, San Marcos, Texas: Texas State University-San Marcos and University of North Texas.
dc.subjectNonlinear boundary conditionsen_US
dc.subjectWeak solutionsen_US
dc.subjectCritical exponenten_US
dc.subjectVariational principleen_US
dc.titleInfinitely many weak solutions for a p-Laplacian equation with nonlinear boundary conditionsen_US
dc.rights.licenseCreative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License.



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