Electron. J. Diff. Eqns., Vol. 2008(2008), No. 98, pp. 1-10.

Existence of solutions for a resonant problem under Landesman-Lazer conditions

Quoc Anh Ngo, Hoang Quoc Toan

Abstract:
This article shows the existence of weak solutions in $W_0^1(\Omega )$ to a class of Dirichlet problems of the form
$$
- \hbox{div}({a({x,\nabla u} )})= \lambda_1 |u|^{p - 2} u
+ f(x,u)-h
$$
in a bounded domain $\Omega$ of $\mathbb{R}^N$. Here a satisfies
$$
|{a({x,\xi } )}| \leq c_0 \big({h_0 (x)+ h_1 (x )|\xi|^{p - 1}}\big)
$$
for all $\xi \in \mathbb{R}^N$, a.e. $x \in \Omega$, $h_0 \in L^{\frac{p}{p - 1}} (\Omega )$, $h_1 \in L_{loc}^1 ( \Omega )$, $h_1(x) \geq 1$ for a.e. x in $x \in \Omega$; $\lambda_1$ is the first eigenvalue for $-\Delta_p$ on $x \in \Omega$ with zero Dirichlet boundary condition and g, h satisfy some suitable conditions.

Submitted March 24, 2008. Published July 25, 2008.
Math Subject Classifications: 35J20, 35J60, 58E05.
Key Words: p-Laplacian; Non-uniform; Landesman-Laser type; Divergence form.

Show me the PDF file (235 KB), TEX file, and other files for this article.

Quocc Anh Ngo
Department of Mathematics, College of Science
Vietnam National University, Hanoi, Vietnam
email: bookworm_vn@yahoo.com, nqanh@vnu.edu.vn
Hoang Quoc Toan
Department of Mathematics, College of Science
Vietnam National University, Hanoi, Vietnam
email: hq_toan@yahoo.com

Return to the EJDE web page