Marco Squassina

Existence results for fractional p-Laplacian problems via Morse theory

Abstract We investigate a class of quasi-linear nonlocal problems, including as a particular case semi-linear problems involving the fractional Laplacian and arising in the framework of continuum mechanics, phase transition phenomena, population dynamics and game theory. Under different growth assumptions on the reaction term, we obtain various existence as well as finite multiplicity results by means of variational and topological methods and, in particular, arguments from Morse theory.

Semiclassical states for weakly coupled nonlinear Schrödinger systems

We consider systems of weakly coupled Schrodinger equations with nonconstant potentials and we investigate the existence of nontrivial nonnegative solutions which concentrate around local minima of the potentials. We obtain sufficient and necessary conditions for a sequence of least energy solutions to concentrate.

Stability and instability results for standing waves of quasi-linear Schrödinger equations

We study a class of quasi-linear Schrodinger equations arising in the theory of superfluid film in plasma physics. Using gauge transforms and a derivation process we solve, under some regularity assumptions, the Cauchy problem. Then, by means of variational methods, we study the existence, the orbital stability and instability of standing waves which minimize some associated energy.

Weyl-type laws for fractional p-eigenvalue problems

We prove an asymptotic estimate for the growth of variational eigenvalues of fractional p-Laplacian eigenvalue problems on a smooth bounded domain.

The Brezis–Nirenberg problem for the fractional p-Laplacian

We obtain nontrivial solutions to the Brezis–Nirenberg problem for the fractional p-Laplacian operator, extending some results in the literature for the fractional Laplacian. The quasilinear case presents two serious new difficulties. First an explicit formula for a minimizer in the fractional Sobolev inequality is not available when

On fractional Choquard equations

p \ne 2

CRITICAL AND SUBCRITICAL FRACTIONAL PROBLEMS WITH VANISHING POTENTIALS

p≠2. We get around this difficulty by working with certain asymptotic estimates for minimizers recently obtained in (Brasco et al., Cal. Var. Partial Differ Equations 55:23, 2016). The second difficulty is the lack of a direct sum decomposition suitable for applying the classical linking theorem. We use an abstract linking theorem based on the cohomological index proved in (Yang and Perera, Ann. Sci. Norm. Super. Pisa Cl. Sci. doi:10.2422/2036-2145.201406_004, 2016) to overcome this difficulty.

Soliton dynamics for fractional Schrödinger equations

Abstract We consider a fractional Schrödinger–Poisson system with a general nonlinearity in the subcritical and critical case. The Ambrosetti–Rabinowitz condition is not required. By using a perturbation approach, we prove the existence of positive solutions. Moreover, we study the asymptotics of solutions for a vanishing parameter.