Electronic Journal of Differential Equations,
Vol. 2008(2008), No. 98, pp. 1-10.

Title: Existence of solutions for a resonant problem under
       Landesman-Lazer conditions
  
Authors: Quoc Anh Ngo (Vietnam National Univ., Hanoi, Vietnam)
, Hoang Quoc Toan     (Vietnam National Univ., Hanoi, Vietnam)
 
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 $\Omega$; $\lambda_1$ is the first
eigenvalue for $-\Delta_p$ on $\Omega$ 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.