Entire sign-changing solutions to the fractional critical Schr{ö}dinger equation

Abstract

We consider the fractional critical Schrödinger equation (FCSE). By virtue of the mini-max theory and the concentration compactness principle with the equivariant group action, we obtain the new type of non-radial, sign-changing solutions of (FCSE) in the energy space Ḣs (R N ). The key component is that we use the equivariant group to partion Ḣs (R N ) into several connected components, then combine the concentration compactness argument to show the compactness property of Palais-Smale sequences in each component and obtain many solutions of (FCSE) in Ḣs (R N ). Both the solutions and the argument here are different from those by Garrido, Musso in [19] and by Abreu, Barbosa and Ramirez in [1].

Existence and convergence of solutions to fractional pure critical exponent problems

“…If G = G i and φ = φ i are those given in (2.14) and (2.15), then Theorem 1.3 yields the existence of at least ⌊ N 4 ⌋ non-radial sign-changing solutions to (1.8). For s ∈ (0, 1), this existence result was proved in the recent paper [TXZZ20], for s = 1 it is shown in [Cla16], and for s = 2 it is a particular case of [CSn20, Theorem 1.1]. All these papers follow a strategy based on a symmetric-concentration compactness argument, but at a technical level they have important differences and none of them can be easily extended to guarantee existence of solutions in the whole higher-order range s ∈ (1, ∞).…”