A critical fractional equation with concave–convex power nonlinearities

Abstract

In this work we study the following fractional critical problemwhere Ω ⊂ R n is a regular bounded domain, λ > 0, 0 < s < 1 and n > 2s. Here (−∆) s denotes the fractional Laplace operator defined, up to a normalization factor, byOur main results show the existence and multiplicity of solutions to problem (P λ ) for different values of λ. The dependency on this parameter changes according to whether we consider the concave power case (0 < q < 1) or the convex power case (1 < q < 2 * s −1). These two cases will be treated separately.

Semilinear problems for the fractional laplacian with a singular nonlinearity

“…that is constructed applying Schauder's fixed point theorem, in the same way as the solution of problem (5). By the maximum principle, u n 0, and thus…”

“…The multiplicity behavior in this case is essentially the same as in concave-convex type problems. Firstly, the approach of [9] based on the use of sub/supersolutions method still works in this context (see also [5,6] where this monotonicity techniques are applied in the nonlocal framework). Then, as in [3], we are also able to show, with some technical variations, the existence of a second solution for the problem .D ; ;p / when we add a convex term if > 0 is small enough.…”

“…To find the critical points of J n , it can be used an adaptation of the ideas developed in [1] (see [5] for the details in the non local case). Indeed, let us define the class…”

Multiple positive solutions for nonlinear coupled fractional Laplacian system with critical exponent

“…When f (x, u) = λu q + u 2 * -1 , Barrios, Colorado, Servadei and Soria [18] obtained the existence and multiplicity solutions for system (1) under different conditions of λ.…”

Three nontrivial solutions for nonlinear fractional Laplacian equations

“…While interior regularity of solutions of (1.1) can be handled just as in the unbounded case, boundary regularity and behaviour of solutions (e.g. the Hopf property) came forth as a serious difficulty, which was mostly overcome by means of weighted Hölder-type function spaces (see [5,21,23,33]). Once provided with the appropriate functional formulation, problem (1.1) becomes variational, in the sense that its weak solutions can be detected as critical points of a C 1 energy functional ϕ, defined on a fractional Sobolev space.…”