Equations involving fractional Laplacian operator: Compactness and application

Abstract

Abstract. In this paper, we consider the following problem involving fractional Laplacian operator:where Ω is a smooth bounded domain in. We show that for any sequence of solutions u n of (1) corresponding to ε n ∈ [0, 2 * α − 2), satisfying u n H ≤ C in the Sobolev space H defined in (1.2), u n converges strongly in H provided that N > 6α and λ > 0. An application of this compactness result is that problem (1) possesses infinitely many solutions under the same assumptions.

Infinitely many sign-changing solutions for the Brézis-Nirenberg problem involving the fractional Laplacian

“…In [10], Caffarelli and Silvestre given a new local realization of the fractional Laplacian (−Δ) s by introducing the so-called s-harmonic extension. After that, several authors, using the localization method, have extended some results of the classical elliptic problems to the fractional Laplacian, see for example [2,5,7,13,16,31,34,35,36,24,25,26] and the references therein. In particular, Chang and Wang [16], using the method of invariant sets of descending flow, obtained the existence and multiplicity of nodal solutions for the elliptic equaitons involving the fractional Laplacian (−Δ) s for all s ∈ (0, 1) with subcritical nonlinearities; for the Brézis-Nirenberg type problem involving the fractional Laplacian (1.1), Tan [31] proved the existence of positive solutions with the special case s = 1 2 and Barrios et al [2] studied the general case with 0 < s < 1.…”

“…In particular, Chang and Wang [16], using the method of invariant sets of descending flow, obtained the existence and multiplicity of nodal solutions for the elliptic equaitons involving the fractional Laplacian (−Δ) s for all s ∈ (0, 1) with subcritical nonlinearities; for the Brézis-Nirenberg type problem involving the fractional Laplacian (1.1), Tan [31] proved the existence of positive solutions with the special case s = 1 2 and Barrios et al [2] studied the general case with 0 < s < 1. For any λ > 0, Yan et al [34] proved that problem (1.1) possesses infinitely many solutions by using a compactness result for the subcritical perturbed problem associated to (1.1). In [21] the authors study bifurcation and multiplicity of solutions for the fractional Laplacian with critical exponential nonlinearity using critical point theorem of Bartolo, Benci and Fortunato [3].…”

“…Multiplying the first eigenfunction and integrating both sides, one can easily check that if λ ≥ λ s 1 , any nontrivial solution of (1.1) is sign-changing. Therefore, by the results of [34], to prove Theorem 1.1, it suffices to consider the case of λ ∈ (0, λ s 1 ). Theorem 1.1 extends the result in [23] to the fractional Laplacian.…”

“…We show that, for each λ > 0, this problem has infinitely many sign-changing solutions by using a compactness result obtained in [34] and a combination of invariant sets method and Ljusternik-Schnirelman type minimax method.MSC 2010 : Primary 35J60; Secondary 47J30, 58E05 …”

Existence for fractional Dirichlet boundary value problem under barrier strip conditions

“…After that, the idea was developed to deal with various fractional boundary value problems such as fractional boundary value problem at resonance [5,6,9,10], Caputo fractional derivative problem [41], impulsive problem [7,31,33], nonlocal problem [4], integral boundary value problem [27], variational structure problem [21], fractional p-Laplace problem [15,20,25,26,34,40], fractional lower and upper solution problem [11,12,42], fractional delay problems, [30,32,44], solitons [16], Biological mathematics [17,29,43], etc. However, all above works were obtained with standard Riemann-Liouville or Caputo fractional derivatives.…”