Giovanni Molica Bisci

Infinitely Many Solutions for a Boundary Value Problem with Discontinuous Nonlinearities

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.

A bifurcation result for non-local fractional equations

In the present paper, we consider problems modeled by the following non-local fractional equation

Infinitely many solutions for the stationary Kirchhoff problems involving the fractional p-Laplacian

\left\{\begin{array}{@{}l@{\quad}l@{}}(-\Delta)^{s} u-\lambda u = \mu f(x,u) & {\rm in}\, \Omega,\\[4pt] u = 0 &{\rm in}\, {\mathbb R}^{n}{\setminus} \Omega,\end{array} \right.

Infinitely many solutions for a Dirichlet problem involving the p-Laplacian

where s ∈ (0, 1) is fixed, (-Δ)s is the fractional Laplace operator, λ and μ are real parameters, Ω is an open bounded subset of ℝn, n > 2s, with Lipschitz boundary and f is a function satisfying suitable regularity and growth conditions. A critical point result for differentiable functionals is exploited, in order to prove that the problem admits at least one non-trivial and non-negative (non-positive) solution, provided the parameters λ and μ lie in a suitable range. The existence result obtained in the present paper may be seen as a bifurcation theorem, which extends some results, well known in the classical Laplace setting, to the non-local fractional framework.

SUPERLINEAR NONLOCAL FRACTIONAL PROBLEMS WITH INFINITELY MANY SOLUTIONS

Abstract This article concerns with a class of nonlocal fractional Laplacian problems depending on two real parameters. Our approach is based on variational methods. We establish the existence of three weak solutions via a recent abstract result by Ricceri about nonlocal equations.

Three Weak Solutions for Nonlocal Fractional Equations

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 Caratheodory 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.

Infinitely many weak solutions for a class of quasilinear elliptic systems

Abstract The aim of this paper is to study a class of nonlocal fractional Laplacian equations depending on two real parameters. More precisely, by using an appropriate analytical context on fractional Sobolev spaces due to Servadei and Valdinoci, we establish the existence of three weak solutions for nonlocal fractional problems exploiting an abstract critical point result for smooth functionals. We emphasize that the dependence of the underlying equation from one of the real parameters is not necessarily of affine type.

On sequences of solutions for discrete anisotropic equations

The existence of infinitely many solutions for an autonomous elliptic Dirichlet problem involving the p-Laplacian is investigated. The approach is based on variational methods.