Competition phenomena for elliptic equations involving a general operator in divergence form

In this paper, by using variational methods, we study an elliptic problem involving a general operator in divergence form of $p$-Laplacian type ($p>1$) with a power $u^q$ and a continuous perturbation 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 $u^q$ and the oscillatory term . To be precise, we prove that, when the perturbation oscillates near the origin, the problem admits infinitely many solutions when $q\geq p-1$ and at least a finite number of solutions when $0p-1$. In all these cases we also give some estimates for the $W^{1,p}$ and $L^\infty$--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.