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

Abstract

In this paper, by using variational methods, we study the following elliptic problem [Formula: see text] involving a general operator in divergence form of [Formula: see text]-Laplacian type ([Formula: see text]). In our context, [Formula: see text] is a bounded domain of [Formula: see text], [Formula: see text], with smooth boundary [Formula: see text], [Formula: see text] is a continuous function with potential [Formula: see text], [Formula: see text] is a real parameter, [Formula: see text] is allowed to be indefinite in sign, [Formula: see text] and [Formula: see text] 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 [Formula: see text] and the oscillatory term [Formula: see text]. To be precise, we prove that, when [Formula: see text] oscillates near the origin, the problem admits infinitely many solutions when [Formula: see text] and at least a finite number of solutions when [Formula: see text]. While, when [Formula: see text] oscillates at infinity, the converse holds true, that is, there are infinitely many solutions if [Formula: see text], and at least a finite number of solutions if [Formula: see text]. In all these cases, we also give some estimates for the [Formula: see text] and [Formula: see text]-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 [Formula: see text]-Laplacian or even to more general differential operators.