Generalized Solutions of Nonlocal Elliptic Problems

Generalized Solutions of Nonlocal Elliptic Problems

An elliptic equation of order 2m with general nonlocal boundary-value conditions, in a plane bounded domain G with piecewise smooth boundary, is considered. Generalized solutions belonging to the Sobolev space Wm 2 (G) are studied. The Fredholm property of the unbounded operator corresponding to the elliptic equation, acting on L2(G), and defined for functions from the space Wm 2 (G) that satis...

Nonlocal Elliptic Boundary Value Problems

Existence, Uniqueness and Multiplicity of Positive Solutions for Some Nonlocal Singular Elliptic Problems

In this article, using the sub-supersolution method and Rabinowitztype global bifurcation theory, we prove some results on existence, uniqueness and multiplicity of positive solutions for some singular nonlocal elliptic problems.

On the Uniqueness of Solutions of a Nonlocal Elliptic System

We consider the following elliptic system with fractional Laplacian −(−∆)su = uv, −(−∆)sv = vu, u, v > 0 on R, where s ∈ (0, 1) and (−∆)s is the s-Lapalcian. We first prove that all positive solutions must have polynomial bound. Then we use the Almgren monotonicity formula to perform a blown-down analysis. Finally we use the method of moving planes to prove the uniqueness of the one dimensional...

Sharp boundary behaviour of solutions to semilinear nonlocal elliptic equations

We investigate quantitative properties of nonnegative solutions u(x) ≥ 0 to the semilinear diffusion equation Lu = f(u), posed in a bounded domain Ω ⊂ R with appropriate homogeneous Dirichlet or outer boundary conditions. The operator L may belong to a quite general class of linear operators that include the standard Laplacian, the two most common definitions of the fractional Laplacian (−∆) (0...