Abstract: In this paper, by using the variational methods, the authors consider the solvability of the equation -△_p u=λa(x)(u⁺)^p-1-μa(x)(u^-)^p-1+f(x, u), u∈
W₀^1,p(\Omega) in the cases of (λ, μ)\not\in ∑_p and (λ, μ) ∈ ∑_p, where \Omega is a bounded domain in R^N(N≥3) with smooth boundary $\partial\Omega$, ∑_p is the Fucik spectrum of the equation -△_p u=α a(x)(u⁺)^p-1-βa(x)(u^-)^p-1, u∈ W₀^1,p(\Omega), the weight function a(x) ∈ L^r(\Omega)(r≥ N/p) with a(x)> 0 a.e. in \Omega, f satisfies some conditions to be specified.

Key words: p-Laplacian, Fucik spectrum, Resonance, Nonresonance, Variational methods