Remarks on the Nonlocal Dirichlet Problem

We study translation-invariant integrodifferential operators that generate Lévy processes. First, we investigate different notions of what a solution to a nonlocal Dirichlet problem is and we provide the classical representation formula for distributional solutions. Second, we study the question under which assumptions distributional solutions are twice differentiable in the classical sense. Sufficient conditions and counterexamples are provided.

Existence of Small-Energy Solutions to Nonlocal Schrödinger-Type Equations for Integrodifferential Operators in ℝN

We are concerned with the following elliptic equations: ( − Δ ) p , K s u + V ( x ) | u | p − 2 u = λ f ( x , u ) in R N , where ( − Δ ) p , K s is the nonlocal integrodifferential equation with 0 < s < 1 < p < + ∞ , s p < N the potential function V : R N → ( 0 , ∞ ) is continuous, and f : R N × R → R satisfies a Carathéodory condition. The present paper is devoted to the study of the L ∞ -bound of solutions to the above problem by employing De Giorgi’s iteration method and the localization method. Using this, we provide a sequence of infinitely many small-energy solutions whose L ∞ -norms converge to zero. The main tools were the modified functional method and the dual version of the fountain theorem, which is a generalization of the symmetric mountain-pass theorem.

Relations of the Algebra of Polynomial Integrodifferential Operators

We revisit the algebra of polynomial integrodifferential operators and we give a generalization of its relations. Generators of prime ideals of height 1, of the maximal ideal, and of the smallest ideal of In are discussed. A technique for obtaining a generating set for a given two-sided ideal of In is also discussed.

Multiplicity of Nontrivial Solutions for a Class of Nonlocal Elliptic Operators Systems of Kirchhoff Type

We investigate the existence and multiplicity of nontrivial solutions for a Kirchhoff type problem involving the nonlocal integrodifferential operators with homogeneous Dirichlet boundary conditions. The main tool used for obtaining our result is Morse theory.