Existence and concentration result for the fractional Schrödinger equations with critical nonlinearities

Existence and concentration of positive ground state solutions for nonlinear fractional Schrödinger‐Poisson system with critical growth

“…One is the L ∞ ‐estimate, owing to the work of Dipierro et al; similarly, we can get the L ∞ ‐estimate. The other is the decay estimate of solutions; with the help of previous works,() we can establish the decay estimate at infinity. Besides above, the novelty of this paper is that we add a potential K ( x ) on the nonlinearity term f ( t ); this will need more careful analysis.…”

“…Proof We borrow some ideas of the proof of theorem 1.1 in He and Zou to give the proof of Lemma . By Lemmas and 4.3 in Felmer et al, by scaling, there exists a continuous function W such that and for some suitable R > 0.…”

“…By adapting some ideas of Benci and Cerami and Benci et al and using the Ljusternick‐Schnirelmann theory, the authors obtained the multiplicity of positive solutions that concentrate on the minima of V ( x ) as ϵ →0. Of course, recently, these methods have been successfully applied to other many problems, such as Schrödinger‐Poisson system, fractional Schrödinger equations, p ‐Laplacian problem, Kirchhoff‐type problems, and the references therein. In one study, by using the methods mentioned before, Liu and Zhang proved the existence and concentration of positive ground state solution for problem when K ( x )≡1 and Q ( x )≡1, but they have not discuss the decay of solutions.…”

Concentration phenomena for a fractional Schrödinger‐Kirchhoff type equation

“…In the last years, the concentration of positive solutions to (4) has attracted the attention of many mathematicians. 20,[28][29][30][31][32][33] In particular, Alves and Miyagaki 28 used the penalization method to study the concentration phenomenon of positive solutions for fractional Schrödinger Equation 4 when V has a local minimum and f is subcritical. He and Zou 33 investigated the relation between the number of positive solutions of (4) with f (u) = g(u) + u 2 * s −1 , where g is subcritical, and the topology of the set where the potential V attains its minima.…”

“…20,[28][29][30][31][32][33] In particular, Alves and Miyagaki 28 used the penalization method to study the concentration phenomenon of positive solutions for fractional Schrödinger Equation 4 when V has a local minimum and f is subcritical. He and Zou 33 investigated the relation between the number of positive solutions of (4) with f (u) = g(u) + u 2 * s −1 , where g is subcritical, and the topology of the set where the potential V attains its minima. In Ambrosio, 29 the first author complemented the results in Alves and Miyagaki 28 and He and Zou 33 dealing with the multiplicity and concentration of solutions in the subcritical and supercritical cases.…”