Multiplicity of radial and nonradial solutions to equations with fractional operators

Abstract

In this paper, we study the existence of radial and nonradial solutions to the scalar field equations with fractional operators. For radial solutions, we prove the existence of infinitely many solutions under N ≥ 2. We also show the existence of least energy solution (with the Pohozaev identity) and its mountain pass characterization. For nonradial solutions, we prove the existence of at least one nonradial solution under N ≥ 4 and infinitely many nonradial solutions under either N = 4 or N ≥ 6. We treat both of the zero mass and the positive mass cases.

A note on deformation argument for $L^2$ constraint problem

“…We also note that a similar approach was taken for nonlinear scalar field equations in Hirata, Ikoma and Tanaka [17] successfully and they gave another proof of the results in [9,10,11]. We also refer to [3,12,13,14,18,19,24] for other applications of generation of Palais-Smale sequences with an extra property. Recently Bartsch and Soave [7] (c.f.…”

Uniqueness of entire ground states for the fractional plasma problem

“…Proof The existence of a weak energy radially decreasing solution to (1.1) could be established by a variation of [31,Theorem 1.3] as it can be reached by proving the existence of radial critical points to the energy functional…”

“…The results in [31] are given for N ≥ 2. Here we provide an alternative variational proof of existence that is well-suited to any dimension N ≥ 1 (recalling that s < 1/2 if N = 1).…”

“…The construction of ground state solutions (by means of critical point theory) for more general subcritical nonlinearities than (1.1) is found in [31]. On the other hand, existence of ground states for the equation (− ) s u = u p with p ≥ (N + 2s)/(N − 2s) is shown in [20, Section 6] and [3] along with a precise decay rate (the case s = 1 is contained in [37,Theorem 9.1]).…”

Deformation Argument under PSP Condition and Applications

“…Such a strategy turned out to be useful for various problems with suitable scaling properties. See [23] for an application for nonlinear Choquard equations, [2,[18][19][20] for fractional scalar field equations, [9] for FitzHugh-Nagumo elliptic systems, [8] for nonlinear elliptic equations in strip-like domains, [3] for nonlinear Schrödiger-Maxwell systems, [4] for nonlinear eigenvalue problems.…”