Infinitely many weak solutions for a p-Laplacian equation with nonlinear boundary conditions
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Date
2007-06-15
Authors
Zhao, Ji-Hong
Zhao, Peihao
Journal Title
Journal ISSN
Volume Title
Publisher
Texas State University-San Marcos, Department of Mathematics
Abstract
We study the following quasilinear problem with nonlinear boundary conditions
-Δpu + 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.
Description
Keywords
p-Laplacian, Nonlinear boundary conditions, Weak solutions, Critical exponent, Variational principle
Citation
Zhao, J. H., & Zhao, P. H. (2007). Infinitely many weak solutions for a p-Laplacian equation with nonlinear boundary conditions. <i>Electronic Journal of Differential Equations, 2007</i>(90), pp. 1-14.
Rights
Attribution 4.0 International