Nonnegative Entire Bounded Solutions to some Semilinear Equations 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...

Asymptotic Spatial Patterns and Entire Solutions of Semilinear Elliptic Equations

(2) ε∆uε + f(uε) = 0, x ∈ Ω, Bu = 0, x ∈ ∂Ω, where ε > 0 is a small parameter, Ω is a smooth bounded domain in Rn, and Bu is an appropriate boundary condition. The connection of (1) and (2) are made by a typical technique called blowup method. Suppose that {uε} is a family of solutions of (2). The simplest setup of the blowup method is to choose Pε ∈ Ω, and define vε(y) = uε(εy + Pε), for y ∈ Ω...

Singular Solutions for some Semilinear Elliptic Equations

We are concerned with the behavior of u near x = O. There are two distinct cases: 1) When p >= N / ( N -2) and (N ~ 3) it has been shown by BR~ZIS & V~RON [9] that u must be smooth at 0 (See also BARAS & PIERRE [1] for a different proof). In other words, isolated singularities are removable. 2) When 1-< p < N / ( N 2) there are solutions of (1) with a singularity at x ---0. Moreover all singula...

Semilinear fractional elliptic equations with gradient nonlinearity involving measures

We study the existence of solutions to the fractional elliptic equation (E1) (−∆)u + ǫg(|∇u|) = ν in a bounded regular domain Ω of R (N ≥ 2), subject to the condition (E2) u = 0 in Ω, where ǫ = 1 or −1, (−∆) denotes the fractional Laplacian with α ∈ (1/2, 1), ν is a Radon measure and g : R+ 7→ R+ is a continuous function. We prove the existence of weak solutions for problem (E1)-(E2) when g is ...

Entire Bounded Solutions for a Class of Quasilinear Elliptic Equations