LAUSR

In this paper, by using variational methods, we study the following elliptic problem involving a general operator in divergence form of p-Laplacian type (p > 1). In our context, Ω is a bounded domain of ℝN, N ≥ 3, with smooth boundary ∂Ω, A is a continuous function with potential a, λ is a real parameter, β ∈ L∞(Ω) is allowed to be indefinite in sign, q > 0 and f : [0, + ∞) → ℝ is a continuous function oscillating near the origin or at infinity. Through variational and topological methods, we show that the number of solutions of the problem is influenced by the competition between the power uq and the oscillatory term f. To be precise, we prove that, when f oscillates near the origin, the problem admits infinitely many solutions when q ≥ p - 1 and at least a finite number of solutions when 0 p - 1. In all these cases, we also give some estimates for the W1, p and L∞-norm of the solutions. The results presented here extend some recent contributions obtained for equations driven by the Laplace operator, to the case of the p-Laplacian or even to more general differential operators.