We consider the equation
−÷ (a(x,|∇u|)∇u) = λ|u|p−2u (whose special case a(x,t) = tp−2 is the p-Laplace
equation) on a bounded domain Ω ⊂ ℝN with C2 boundary, with null boundary
condition. We prove that there are λ ∈ ℝ for which the equation has a nontrivial
solution. As an application, by variational methods, we present the existence of a
positive solution to −÷ (a(x,|∇u|)∇u) = f(x,u) in Ω, where f is asymptotically
(p−1)-linear near zero and ∞, considering the nonresonant, resonant, and
doubly resonant cases. We show that, generally, the spectrum of the operator
−÷ (a(x,|∇u|)∇u) on W01,p(Ω) is not discrete.