Rocky Mountain Journal of Mathematics

Eigenvalues of some $p(x)$-biharmonic problems under Neumann boundary conditions

Mounir Hsini, Nawal Irzi, and Khaled Kefi

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In this paper, we study the following $p(x)$-biharmonic problem in Sobolev spaces with variable exponents \begin{equation} \begin{cases} \triangle ^{2}_{p(x)}u=\lambda ({\partial F } (x,u)/{\partial u}) & x\in \Omega , \\ {\partial u}/{\partial n}=0 & x\in \partial \Omega ,\\ {\partial }(|\triangle u|^{p(x)-2}\triangle u)/{\partial n} =a(x)|u|^{p(x)-2}u & x\in \partial \Omega . \end{cases} \end{equation} By means of the variational approach and Ekeland's principle, we establish that the above problem admits a nontrivial weak solution under appropriate conditions.

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Rocky Mountain J. Math., Volume 48, Number 8 (2018), 2543-2558.

First available in Project Euclid: 30 December 2018

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Primary: 35D05 35D30: Weak solutions 35J58: Boundary value problems for higher-order elliptic systems 35J60: Nonlinear elliptic equations 35J65: Nonlinear boundary value problems for linear elliptic equations

$p(x)$-biharmonic operator Ekeland's variational principle generalized Sobolev spaces weak solution


Hsini, Mounir; Irzi, Nawal; Kefi, Khaled. Eigenvalues of some $p(x)$-biharmonic problems under Neumann boundary conditions. Rocky Mountain J. Math. 48 (2018), no. 8, 2543--2558. doi:10.1216/RMJ-2018-48-8-2543.

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