Bound state solutions for the supercritical fractional Schrödinger equation

Abstract

We prove the existence of positive solutions for the supercritical nonlinear fractionalfor s ∈ (0, 1), n > 2s. We show that if V (x) = o(|x| −2s ) as |x| → +∞, then for p > n+2s−1 n−2s−1 , this problem admits a continuum of solutions. More generally, for p > n+2s n−2s , conditions for solvability are also provided. This result is the extension of the work by Davila, Del Pino, Musso and Wei to the fractional case. Our main contributions are: the existence of a smooth, radially symmetric, entire solution of (−∆) s w = w p in R n , and the analysis of its properties. The difficulty here is the lack of phase-plane analysis for a nonlocal ODE; instead we use conformal geometry methods together with Schaaf's argument as in the paper by Ao, Chan, DelaTorre, Fontelos, González and Wei on the singular fractional Yamabe problem.

Construct new type solutions for the fractional Schrödinger equation

“…For the supercritical case, Ao et al in [4] proved the existence of bound state solutions for (1.4). When the potential and nonlinearity satisfy certain conditions, Bisci and Rădulescu in [30] studied the existence of multiple ground state solutions for the following problem:…”

ODE Methods in Non-Local Equations

“…Here we have used e(ρ) → 1 as ρ → 0, and the second equation in (6.2). Similarly, it holds that I 2 = − 1 2 e 2 ρ 1−2γ (∂ t W) 2 Now we provide another Pohožaev type identity based on the variation of constants formula. We discuss first the simpler case where all τ j =0, then the general case.…”

“…where 2 . In order to relate to our setting, first we need to shift the information from r=∞ to the origin, so we set t = logr (note the sign change with respect to the above!).…”

“…The fractional Hardy inequality ( [8,30,39,41,42,50]) asserts the non-negativity of such operator up to κ = Λ n,γ , hence whenever 6) and this distinguishes the stable/unstable cases (see Definition 3.1). Nonlinear Schrödinger equations with fractional Laplacian have received a lot of attention recently (see, for instance, [2,21,25]), while ground state solutions for non-local problems have been considered in the papers [28,29].…”

Fast and slow decay solutions for supercritical fractional elliptic problems in exterior domains

“…In particular, fractional elliptic problems have been extensively studied. See for example [8,9,15] for subcritical case, and [1,2,18] for critical exponent, [3] for the supercritical case and the reference therein.…”