Uniqueness of Positive Solutions for Fractional Schrödinger Equations with General Nonlinearities

2024/01/05 by Xinyu Li, Linjie Song, Li, Xinyu +1
Mathematics · Computer Science · #Nonlinear Partial Differential Equations #Advanced Mathematical Modeling in Engineering #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2401.02795

Abstract

In 2013, R.L. Frank and E. Lenzmann [R.L. Frank, E. Lenzmann, Uniqueness of non-linear ground states for fractional Laplacians in ℝ, Acta Math. 210 (2) (2013) 261-318] study the following problem: (-Δ)^su + u = u^p-1 in ℝ^N, where s ∈ (0,1), N = 1, p ∈ (2,2_s^*), and 2_s^* is the critical fractional Sobolev exponent. They proved that the ground state is unique (up to translations). Then in 2016, they, together with L. Silvestre [R.L. Frank, E. Lenzmann, L. Silvestre, Uniqueness of radial solutions for the fractional Laplacian, Comm. Pure Appl. Math. 69 (9) (2016) 1671-1726] showed similar uniqueness results for high dimensions (N \geqslant 2), in which they proposed a challenging open problem to extend their results about non-degeneracy and uniqueness of ground states to nonlinearities f(u) beyond the pure-power case. To the best of our knowledge, this question is still unresolved so far. In this paper, we aim to give a full affirmative answer to this open issue for a large class of convex nonlinearities. In fact, we prove the uniqueness of the positive solution.

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