Detection of arbitrarily many solutions for perturbed elliptic problems involving oscillatory terms

We propose a direct approach for detecting arbitrarily many solutions for perturbed elliptic problems involving oscillatory terms. Although the method works in various frameworks, we illustrate it on the problemequation(Pε){−Δu+u=Q(x)[f(u)+εg(u)],x∈RN,N⩾2,u⩾0,u(x)→0as |x|→∞, where Q:RN→R is a radial, positive potential, f:[0,∞)→R is a continuous nonlinearity which oscillates near the origin or at infinity and g:[0,∞)→R is any arbitrarily continuous function with g(0)=0g(0)=0. Our aim is to prove that: (a) the unperturbed problem (P0)(P0), i.e. ε=0ε=0 in (Pε), has infinitely many distinct solutions; (b) the number of distinct solutions for (Pε) becomes greater and greater whenever |ε||ε| is smaller and smaller. In fact, our method surprisingly shows that (a) and (b) are equivalent   in the sense that they are deducible from each other. Various properties of the solutions are also described in L∞L∞- and H1H1-norms. Our method is variational and a specific construction enforces the use of the principle of symmetric criticality for non-smooth Szulkin-type functionals.