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

This article shows the existence of weak solutions in $W_0^1(Omega )$ to a class of Dirichlet problems of the form $$ - hbox{div}({a({x, abla u} )})= lambda_1 |u|^{p - 2} u + f(x,u)-h $$ in a bounded domain $Omega$ of $mathbb{R}^N$. Here $a$ satisfies $$ |{a({x,xi } )}| leq c_0 ig({h_0 (x)+ h_1 (x )|xi|^{p - 1}}ig) $$ 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.