Two nontrivial solutions for an elliptic problem involving some nonlocal integro-differential operators

Renormalized and entropy solutions for the fractional p-Laplacian parabolic equation with L^1 data

“…For the convenience of the readers, we recall some definitions and basic properties of the fractional Sobolev spaces, in which main results can be found in [22,29,32,41,42] and the references therein.…”

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.…”

Ground state and nodal solutions for fractional Kirchhoff equation with pure critical growth nonlinearity

“…Some of these additional difficulties were overcome in [12,13]. More studies on this integro-differential operator equation, for examples, positive, negative, sign-changing or ground state solutions, we refer the papers [4,5,9,16] and their references. In [1], the authors considered the obstacle problems for L K about the higher regularity of free boundaries.…”