One-sided multiple resonance for quasilinear elliptic partial differential equations

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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.