The existence of infinitely many solutions for a Sturm-Liouville boundary value problem, under an appropriate oscillating behavior of the possibly discontinuous nonlinear term, is obtained. Several special cases and consequences are pointed out and some examples are presented. The technical approach is mainly based on a result of infinitely many critical points for locally Lipschitz functions.

In this paper we study the existence of infinitely many weak solutions for equations driven by nonlocal integrodifferential operators with homogeneous Dirichlet boundary conditions. A model for these operators is given by the fractional Laplacian where s ∈ (0, 1) is fixed. We consider different superlinear growth assumptions on the nonlinearity, starting from the well-known Ambrosetti–Rabinowitz condition. In this framework we obtain three different results about the existence of infinitely many weak solutions for the problem under consideration, by using the Fountain Theorem. All these theorems extend some classical results for semilinear Laplacian equations to the nonlocal fractional setting.

The aim of this paper is to establish the multiplicity of weak solutions for a Kirchhoff-type problem driven by a fractional p -Laplacian operator with homogeneous Dirichlet boundary conditions: where is an open bounded subset of with Lipshcitz boundary , is the fractional p -Laplacian operator with 0 < s < 1 < p < N such that sp < N , M is a continuous function and f is a Carathéodory function satisfying the Ambrosetti–Rabinowitz-type condition. When f satisfies the suplinear growth condition, we obtain the existence of a sequence of nontrivial solutions by using the symmetric mountain pass theorem; when f satisfies the sublinear growth condition, we obtain infinitely many pairs of nontrivial solutions by applying the Krasnoselskii genus theory. Our results cover the degenerate case in the fractional setting: the Kirchhoff function M can be zero at zero.