## International Journal of Differential Equations

### Multiplicity Results for the $p(x)$-Laplacian Equation with Singular Nonlinearities and Nonlinear Neumann Boundary Condition

#### Abstract

We investigate the singular Neumann problem involving the $p(x)$-Laplace operator: $({P}_{\lambda })\{-{\mathrm{\Delta }}_{p(x)}u+|u{|}^{p(x)-\mathrm{2}}u$  $=\mathrm{1}/{u}^{\delta (x)}+f(x,u)$, in  $\mathrm{\Omega }; u>\mathrm{0}, \text{in} \mathrm{\Omega }; {|\nabla u|}^{p(x)-\mathrm{2}}\partial u/\partial \nu =\lambda {u}^{q(x)}, \text{on} \partial \mathrm{\Omega }\}$, where $\mathrm{\Omega }\subset {\mathbb{R}}^{N}(N\ge \mathrm{2})$ is a bounded domain with ${C}^{\mathrm{2}}$ boundary, $\lambda$ is a positive parameter, and $p(x),$$q(x),$$\delta (x)$, and $f(x,u)$ are assumed to satisfy assumptions (H0)(H5) in the Introduction. Using some variational techniques, we show the existence of a number $\mathrm{\Lambda }\in (\mathrm{0},\mathrm{\infty })$ such that problem $({P}_{\lambda })$ has two solutions for $\lambda \in (\mathrm{0},\mathrm{\Lambda }),$ one solution for $\lambda =\mathrm{\Lambda }$, and no solutions for $\lambda >\mathrm{\Lambda }$.

#### Article information

Source
Int. J. Differ. Equ., Volume 2016 (2016), Article ID 3149482, 14 pages.

Dates
Received: 5 April 2016
Accepted: 22 June 2016
First available in Project Euclid: 21 December 2016

Permanent link to this document
https://projecteuclid.org/euclid.ijde/1482352600

Digital Object Identifier
doi:10.1155/2016/3149482

Mathematical Reviews number (MathSciNet)
MR3574266

Zentralblatt MATH identifier
06915917

#### Citation

Saoudi, K.; Kratou, M.; Alsadhan, S. Multiplicity Results for the $p(x)$ -Laplacian Equation with Singular Nonlinearities and Nonlinear Neumann Boundary Condition. Int. J. Differ. Equ. 2016 (2016), Article ID 3149482, 14 pages. doi:10.1155/2016/3149482. https://projecteuclid.org/euclid.ijde/1482352600

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