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.

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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

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