Concentration phenomena for a fractional Schrödinger‐Kirchhoff type equation

Abstract

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

On a fractional magnetic Schrödinger equation inRwith exponential critical growth

“…then (3.26) yields that (3.25) holds true. However, the validity of the limit in (3.28) can be obtained arguing as in the proof of Lemma 2.8 observing that in this case we have to use the boundedness of (u n ) in H 1/2 ε (and then the boundedness of (|u n |) in L q (R, R) for all q ∈ [2, ∞)) to get estimates independent of n ∈ N; see also Lemma 3.4 in [11].…”

Multiplicity and Concentration of Solutions for a Fractional Kirchhoff Equation with Magnetic Field and Critical Growth

“…Moreover, we can argue as in Lemma 6.1 in [10] to infer that u ∈ L ∞ (R 3 , R). Since u satisfies [53]) and that u > 0 by maximum…”

“…Liu et al [44] used the monotonicity trick and the profile decomposition, to obtain the existence of ground states to a fractional Kirchhoff equation with critical nonlinearity in low dimension. The author and Isernia in [10] studied the existence and multiplicity via penalization method and the Ljusternik-Schnirelmann category theory for a fractional Kirchhoff equation with subcritical nonlinearities.…”

Concentration phenomena for a class of fractional Kirchhoff equations inRNwith general nonlinearities

“…In the fractional context, after the pioneering work [32], many authors focused on fractional Kirchhoff problems set in bounded domains or in the whole space and involving nonlinearities with subcritical and critical growth; see for instance [10,30,42,45,50] and the references therein for unperturbed problems (that is when ε = 1 in (1.1)), and [9,11,37] for some existence and multiplicity results for perturbed problems (that is when ε > 0 is sufficiently small). On the other hand, when M (t) ≡ 1, equation (1.1) boils down to a nonlinear fractional Schrödinger equation of the type ε 2s (−∆) s u + V (x)u = h(x, u) in R N .…”