Infinitely many solutions of a quasilinear elliptic problem with an oscillatory potential

Let {Omega} be a bounded domain in IR{sup N}, with N {ge} 1, having a smooth boundary {partial_derivative}{Omega}. We denote by A the quasilinear elliptic second order differential operator defined by Au+div(a({vert_bar}{del}{sub u}{vert_bar}{sup 2}){del}{sub u}). We suppose that the function a:[O,+{infinity}{r_arrow}O, +{infinity}] is of class C{sup 1} and satisfies the following ellipticity and growth conditions of Leray-Lions type (cf. e.g. [22]): there are constants {gamma}, {Lambda} > O, K {epsilon} [O,1] and p {epsilon}[1, +{infinity}]such that, for every s > O, {lambda}(K + S){sup p-2} {le} a(s{sup 2}){le}{Lambda} (K+S){sup p-2}({lambda}-1/2) a(s){le}a{prime}(s) s {le}{Gamma} a(s). Hence, we can define, for each s {ge} O, the function A(s) = {integral}{sub O}{sup s} a({xi})d{xi}. Let us consider the Dirichlet problem -Au={mu}(x)g(u) + h(x) in {Omega}, u=O on {partial_derivative}{Omega}, where g: IR {r_arrow} IR is continuous and {mu}, h {epsilon} L{sup {infinity}}({infinity}), with {mu}{sub O} = ess inf{sub {Omega}}{sub {mu}} > O. We also set G(s) = {integral}{sub O}{sup s}g({integral})d{integral}, for all s {epsilon} IR. By a solution of (1.3) we mean a function u {epsilon} W{sub O}{sup 1,p} ({Omega}) {intersection} L{sup {infinity}} ({Omega}) such that {integral}{sub {Omega}} a({vert_bar}{del}{sub u}{vert_bar}{sup 2}){del}{sub u}{del}{sub wdx}= {integral}{sub {Omega}} {mu}g(u)wdx + {integral}{sub {Omega}} hwdx, for every w {epsilon} W{sub O}{sup 1,p}({Omega}), where p is the exponent which appears in (1.1). The aim of this paper is to prove the existence of infinitely many solutions of problem (1.3) when the potential G(s) exhibits an oscillatory behaviour at infinity. 22 refs.