One-sided multiple resonance for quasilinear elliptic partial differential equations

Abstract

In this article, we apply the Galerkin approximation method to obtain an existence theorem of weak solutions for 2mth order quasilinear elliptic partial differential resonance equations −Q(u)+f(x,u) =G on a bounded open connected subset Ω of

in which the nonlinearity f(x,u) has no growth restriction in u in one of the directions (u→ ∞ and u→ −∞ ) and belongs to o(|u| _p−1 ) (p> N/m) in the other, and G∈ [ ^m,p (Ω )]* may satisfy a Landesman–Lazer condition.

Keywords:

• Galerkin approximation method
• Multiple resonance
• 2000 Mathematics Subject Classifications: 35A
• 35J
• 47H

Acknowledgment

This research was supported in part by the National Science Council of R.O.C.