Nodal and multiple solutions of nonlinear problems involving the fractional Laplacian

Multiplicity results for the non-homogeneous fractionalp-Kirchhoff equations with concave–convex nonlinearities

“…This type of operators arises in a quite natural way in many different applications, such as continuum mechanics, phase transition phenomena, population dynamics, minimal surfaces and game theory, as they are the typical outcome of stochastically stabilization of Lévy processes (see [5][6][7][8][9] and the references therein). The literature on fractional operators and their applications is very huge, here we just mention a few, see for example [10][11][12][13][14][15][16][17][18][19][20], especially [21] and [22,23] for two different fractional Laplacian involving concave-convex nonlinearities. On the subject of concave-convex nonlinearities involving the classical Laplacian operator, for instance, we refer the reader to [24].…”

Three nontrivial solutions for nonlinear fractional Laplacian equations

“…So, we can prove existence and multiplicity of such solutions by applying to ϕ several abstract results of critical point theory, such as minimax principles (see [32]) and Morse theory (see [13]). Some results of this type can be found, for instance, in [6,14,17,25,27,29,38].…”

Existence and nonexistence of positive solutions to the fractional equation Δ^α/2 u = –u^γ in bounded domains

“…, it was proved in [8,19] that problem (1) admits a positive solution in the fractional Sobolev space H α 2 (D). The nonexistence result in starshaped domains has been investigated in [14,18].…”

“…In both works, the authors proved that if γ ≥ d+α d−α then problem (1) has no positive bounded solutions. A natural question to ask is how the solutions of (1) in [8,19] behave near the boundary ∂D (bounded or not)? Also, what can be said about the existence of unbounded solutions of (1) for γ ≥ d+α d−α ?…”

“…This work hopefully allows us to better understand how the nonlocal property of the fractional Laplacian influences the boundary behavior of solutions. The approach that we perform in this paper is essentially based on some tools from potential theory and it is completely different from that used in [1,8,14,18,19].…”