Large solutions to elliptic equations involving fractional Laplacian

Existence of positive solutions to elliptic problems involving the fractional Laplacian

Semilinear fractional elliptic equations involving measures

We study the existence of weak solutions to (E) (−∆)u+g(u) = ν in a bounded regular domain Ω in R (N ≥ 2) which vanish in R \Ω, where (−∆) denotes the fractional Laplacian with α ∈ (0, 1), ν is a Radon measure and g is a nondecreasing function satisfying some extra hypotheses. When g satisfies a subcritical integrability condition, we prove the existence and uniqueness of a weak solution for pr...

Saddle-shaped Solutions of Bistable Elliptic Equations Involving the Half-laplacian

We establish existence and qualitative properties of saddle-shaped solutions of the elliptic fractional equation (−∆)u = f(u) in all the space R, where f is of bistable type. These solutions are odd with respect to the Simons cone and even with respect to each coordinate. More precisely, we prove the existence of a saddle-shaped solution in every even dimension 2m, as well as its monotonicity p...

EXISTENCE OF WEAK SOLUTIONS FOR QUASILINEAR ELLIPTIC EQUATIONS INVOLVING THE p-LAPLACIAN

This paper shows the existence of nontrivial weak solutions for the quasilinear elliptic equation − ` ∆pu +∆p(u ) ́ + V (x)|u|p−2u = h(u) in RN . Here V is a positive continuous potential bounded away from zero and h(u) is a nonlinear term of subcritical type. Using minimax methods, we show the existence of a nontrivial solution in C loc (R N ) and then show that it decays to zero at infinity wh...

Multiplicity of Solutions for Non-local Elliptic Equations Driven by Fractional Laplacian

A. We consider the semi-linear elliptic PDEs driven by the fractional Laplacian: { (−∆)su = f (x, u), in Ω, u = 0, in Rn\Ω. By the Mountain Pass Theorem and some other nonlinear analysis methods, the existence and multiplicity of non-trivial solutions for the above equation are established. The validity of the Palais-Smale condition without AmbrosettiRabinowitz condition for non-local el...