Existence and multiplicity of positive solutions for fractional Laplacian systems with nonlinear coupling

Abstract

It is well known that a single nonlinear fractional Schrödinger equation with a potential V (x) may have a positive solution that is concentrated at the nondegenerate minimum point of V (x) as the positive parameter ε sufficiently small (see [11,14,22]). While in this paper, we can find two different positive solutions for weakly coupled fractional Schrödinger systems with two potentials V 1 (x) and V 2 (x) having the same minimum point and these positive solutions are concentrated at this minimum point. In fact that by using the energy estimates, Nehari manifold technique and the Lusternik-Schnirelmann theory of critical points, we obtain the multiplicity results for a class of weakly coupled fractional Schrödinger system. Furthermore, the existence and nonexistence of least energy positive solutions are also explored.

Existence of positive ground state solutions to a nonlinear fractional Schrödinger system with linear couplings

“…(1.6) possesses ground state solutions by using variational methods. For other work about the fractional Laplacian system, we would like to mention [13][14][15][16][17][18] and the references therein.…”

Fractional Schrodinger-Poisson systems with weighted Hardy potential and critical exponent

“…We consider the operator From the mathematical point of view, a reason of interest in potentials of the type Φ( x |x| )/|x| 2s relies in their criticality with respect to the differential operator (−∆) s ; indeed, they have the same homogeneity as the fractional Laplacian (−∆) s , hence they cannot be regarded as a lower order perturbation term. We mention that the operator with singular potentials have been widely studied, see for example [1,13,14,15,17,18,19,20,21,34,36,38,39] and references therein.…”

Bound state positive solutions for a class of elliptic system with Hartree nonlinearity

“…Then by a similar argument to that used in the proof of Lemma 3.1 in [10] and the Ekeland variational principle [14], we can prove the existence of (P S) γi,j (ε) -sequence as follows:…”