We consider a class of parametric Schrödinger equations driven by the fractional p-Laplacian operator and involving continuous positive potentials and nonlinearities with subcritical or critical growth. By using variational methods and Ljusternik-Schnirelmann theory, we study the existence, multiplicity and concentration of positive solutions for small values of the parameter.

Δ) s is the fractional Laplacian, M is a Kirchhoff function, V is a continuous positive potential, and f is a superlinear continuous function with subcritical growth. By using penalization techniques and Ljusternik-Schnirelmann theory, we investigate the relation between the number of positive solutions with the topology of the set where the potential attains its minimum. KEYWORDS fractional Schrödinger-Kirchhoff problem, Ljusternik-Schnirelmann theory, Moser iteration, Nehari manifold, variational methods Math Meth Appl Sci. 2018;41 615-645.wileyonlinelibrary.com/journal/mma

Abstract. In this paper we deal with the following fractional Kirchhoff equationwhere s ∈ (0, 1), N ≥ 2, p > 0, q is a small positive parameter and g : R → R is an odd function satisfying Berestycki-Lions type assumptions. By using minimax arguments, we establish a multiplicity result for the above equation, provided that q is sufficiently small.

In this paper we consider a class of fractional Schrödinger equations with potentials vanishing at infinity. By using a minimization argument and a quantitative Deformation Lemma, we prove the existence of a sign-changing solution.where P.V. stands for the Cauchy principal value and C N,α is a normalizing constant [22]. Here, we assume that V, K : R N → R are continuous functions verifying appropriate hypotheses. More precisely, as in [1], we say (V, K) ∈ K if the following conditions hold:2010 Mathematics Subject Classification. 35A15, 35J60, 35R11, 45G05.

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