Three nontrivial solutions for nonlinear fractional Laplacian equations

Abstract

Abstract. We study a Dirichlet-type boundary value problem for a pseudodifferential equation driven by the fractional Laplacian, proving the existence of three nonzero solutions. When the reaction term is sublinear at infinity, we apply the second deformation theorem and spectral theory. When the reaction term is superlinear at infinity, we apply the mountain pass theorem and Morse theory.

Sobolev versus Hölder minimizers for the degenerate fractional p-Laplacian

“…To conclude we just want to give an example of how our result works, presenting a nonlinear extension of [8,Theorem 3.3]. We make on the reaction f in problem (1.3) the following assumptions:…”

“…Under these assumptions, we prove the following multiplicity result for problem (1.3): Proof. We just sketch the proof, referring to [8] for details. First we introduce two truncated reactions and their primitives, defined for all (x, t) ∈ Ω × R by…”

Existence and multiplicity results for resonant fractional boundary value problems

“…When combined with a non-linear reaction term, the fractional Laplacian gives rise to non-local boundary value problems like (1.1), where the Dirichlet condition is stated on the complementary set Ω c = R N \ Ω, instead of just on ∂Ω, for both formal and intrinsic reasons, see [5,32,33]. The problem admits a weak formulation and can be treated through variational methods and critical point theory: some results in this direction can be found in [1,2,4,10,11,14,16,17,22,24,26,28,32,35,36] (see also [21,23] for problems involving the nonlinear corresponding operator, namely the fractional p-Laplacian). Just as in the classical case, knowing the asymptotic behavior (sub-linear, linear, or super-linear) of the reaction term f (x, ·) gives precious information about the existence and multiplicity of solutions.…”

“…Just as in the classical case, knowing the asymptotic behavior (sub-linear, linear, or super-linear) of the reaction term f (x, ·) gives precious information about the existence and multiplicity of solutions. While in [14] the cases of sub-or super-linear reactions at infinity were considered, here we want to focus on the case when f (x, ·) is asymptotically linear at infinity, with a positive limit slope. The main tool in dealing with such reactions is a comparison with the corresponding eigenvalue problem (a similar approach was recently applied to a semilinear Robin problem, see [29]).…”

Existence of Three Positive Solutions for a Nonlocal Singular Dirichlet Boundary Problem

“…They established existence, regularity and bifurcation results using the framework of weighted spaces. Recently, [16] has established the existence of three non-zero solutions for a Dirichlet type boundary value problem involving the fractional Laplacian. But the study of three solutions for singular nonlocal problems was completely open till now.…”