Electron. J. Diff. Eqns., Vol. 2008(2008), No. 165, pp. 1-6.

Multiple solutions for quasilinear elliptic problems with nonlinear boundary conditions

Nguyen Thanh Chung

Abstract:
Using a recent result by Bonanno [2], we obtain a multiplicity result for the quasilinear elliptic problem
$$\displaylines{ - \Delta_p u + |u|^{p-2}u = \lambda f(u) \quad \hbox{in } \Omega, \cr |\nabla u|^{p-2} \frac{\partial u}{\partial \nu} = \mu g(u) \quad \hbox{on } \partial\Omega, }$$
where $\Omega$ is a bounded domain in $\mathbb R^N$, $N \geq 3$ with smooth boundary $\partial\Omega$, $\frac{\partial}{\partial\nu}$ is the outer unit normal derivative, the functions $f, g$ are $(p-1)$-sublinear at infinity ($1<p<N$), $\lambda$ and $\mu$ are positive parameters.

Submitted October 20, 2008. Published December 23, 2008.
Math Subject Classifications: 35J65, 35J20.
Key Words: Multiple solutions; quasilinear elliptic problems; nonlinear boundary conditions

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Nguyen Thanh Chung
Department of Mathematics and Informatics
Quang Binh University
312 Ly Thuong Kiet, Dong Hoi, Quang Binh, Vietnam
email: ntchung82@yahoo.com

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