Class Of Nonlocal Elliptic Equations Research Articles

On BMO and Hardy regularity estimates for a class of non-local elliptic equations

Let $\sigma \in (0,\,2)$, $\chi ^{(\sigma )}(y):={\mathbf 1}_{\sigma \in (1,2)}+{\mathbf 1}_{\sigma =1} {\mathbf 1}_{y\in B(\mathbf {0},\,1)}$, where $\mathbf {0}$ denotes the origin of $\mathbb {R}^n$, and $a$ be a non-negative and bounded measurable function on $\mathbb {R}^n$. In this paper, we obtain the boundedness of the non-local elliptic operator \[ Lu(x):=\int_{\mathbb{R}^n}\left[u(x+y)-u(x)-\chi^{(\sigma)}(y)y\cdot\nabla u(x)\right]a(y)\,\frac{{\rm d}y}{|y|^{n+\sigma}} \]from the Sobolev space based on $\mathrm {BMO}(\mathbb {R}^n)\cap (\bigcup _{p\in (1,\infty )}L^p(\mathbb {R}^n))$ to the space $\mathrm {BMO}(\mathbb {R}^n)$, and from the Sobolev space based on the Hardy space $H^1(\mathbb {R}^n)$ to $H^1(\mathbb {R}^n)$. Moreover, for any $\lambda \in (0,\,\infty )$, we also obtain the unique solvability of the non-local elliptic equation $Lu-\lambda u=f$ in $\mathbb {R}^n$, with $f\in \mathrm {BMO}(\mathbb {R}^n)\cap (\bigcup _{p\in (1,\infty )}L^p(\mathbb {R}^n))$ or $H^1(\mathbb {R}^n)$, in the Sobolev space based on $\mathrm {BMO}(\mathbb {R}^n)$ or $H^1(\mathbb {R}^n)$. The boundedness and unique solvability results given in this paper are further devolvement for the corresponding results in the scale of the Lebesgue space $L^p(\mathbb {R}^n)$ with $p\in (1,\,\infty )$, established by H. Dong and D. Kim [J. Funct. Anal. 262 (2012), 1166–1199], in the endpoint cases of $p=1$ and $p=\infty$.

Existence and Multiplicity of Solutions for a Biharmonic Equation with p(x)-Kirchhoff Type

Based on the basic theory and critical point theory of variable exponential Lebesgue Sobolev space, this paper investigates the existence and multiplicity of solutions for a class of nonlocal elliptic equations with Navier boundary value conditions when (AR) condition does not hold and improves or generalizes the original conclusions.

Formula omitted] regularity theory for a class of nonlocal elliptic equations

In this paper, we study the regularity of weak solutions to a class of nonlocal elliptic equations in Bessel potential spaces Hs,p. Our main results can be seen as an extension of the well-known W1,p regularity theory for local second-order elliptic equations in divergence form to the nonlocal setting.

The Dirichlet problem for nonlocal elliptic equations

We study a class of nonlocal elliptic equations on a bounded domain . For (), by the Lax–Milgram theorem and the De Giorgi iteration, we prove the existence and uniqueness of solution. Furthermore, we investigate the existence of densities for measure-valued solutions to nonhomogeneous measure-valued nonlocal elliptic equations.

Positive solution for a class of nonlocal elliptic equations

In this paper, we devote ourselves to investigating the existence of positive solution for a class of nonlocal elliptic equations. Our approach is based on the fixed point index theory.

On a class of non-local elliptic equations with asymptotically linear term

We consider the nonlinear elliptic PDE driven by the fractional Laplacian with asymptotically linear term. Some results regarding existence and multiplicity of non-trivial solutions are obtained. More precisely, information about multiple non-trivial solutions is given under some hypotheses of asymptotically linear condition; non-local elliptic equations with combined nonlinearities are also studied, and some results of local existence and global existence are obtained. Finally, an $L^{∞}$ regularity result is also given in the appendix, using the De Giorgi-Stampacchia iteration method.

Properties of unique positive solution for a class of nonlocal semilinear elliptic equation

We study a class of nonlocal elliptic equations $$ -M\bigg(\int_{\Omega}|u|^{\gamma}dx\bigg)\Delta u=\lambda f(x,u) $$ with the Dirichlet boundary conditions in bounded domain. Under suitable assumptions on $M$ and the nonlinear term $f$, the existence and new properties of a unique positive solutions are obtained via a monotone operator method and a mixed monotone operator method.

On the Convergence of a Class of Nonlocal Elliptic Equations and Related Optimal Design Problems

A convergence result for a nonlocal differential equation problem is proved. As a by-product, some results about the convergence for a type of nonlocal optimal design are given. Since these problems give rise to local design problems in the limit, different results on classical existence are obtained as well. Concerning the nonlocal formulation, the state equation is of nonlocal elliptic type and the cost functional we analyze includes, among other cases, an approximation of the square of the gradient.

Eigenvalues, global bifurcation and positive solutions for a class of nonlocal elliptic equations

In this paper, we shall study global bifurcation phenomenon for the following Kirchhoff type problem: \begin{equation*} \begin{cases} \displaystyle -\bigg(a+b\int_\Omega \vert \nabla u\vert^2dx\bigg)\Delta u=\lambda u+h(x,u,\lambda) &\text{in } \Omega,\\ u=0 &\text{on }\Omega. \end{cases} \end{equation*} Under some natural hypotheses on $h$, we show that $(a\lambda_1,0)$ is a bifurcation point of the above problem. As an application of the above result, we shall determine the interval of $\lambda$, in which there exist positive solutions for the above problem with $h(x,u;\lambda)=\lambda f(x,u)-\lambda u$, where $f$ is asymptotically linear at zero and asymptotically 3-linear at infinity. To study global structure of bifurcation branch, we also establish some properties of the first eigenvalue for a nonlocal eigenvalue problem. Moreover, we provide a positive answer to an open problem involving the case $a=0$.

A Hölder Regularity Theory for a Class of Non-Local Elliptic Equations Related to Subordinate Brownian Motions

In this article we present the existence, uniqueness, and Holder regularity of solutions for the non-local elliptic equations of the type $$Lu -\lambda u = f \quad \text{in} \quad \mathbb{R}^{d}, $$ where $$Lu(x)={\int}_{\mathbb{R}^{d}}(u(x+y)-u(x)-y \cdot \nabla u(x) \chi (y) )\, a(y)J(y)dy. $$ Here χ(y) is a suitable indicator function, J(y)d y is a rotationally invariant Levy measure on ℝ d (i.e. \({\int }_{\mathbb {R}^{d}}\left (1\wedge |y|^{2}\right )J(y)dy<\infty \)), and a(y) is an only measurable function with positive lower and upper bounds.

The existence of multiple positive solutions for a class of non-local elliptic problem in RN

This paper is concerned with the existence of at least two positive solutions for a class of non-local elliptic equations in . Using variational methods and the decomposition of the Nehari manifold, we show multiple existence of positive solutions.

Schauder estimates for a class of non-local elliptic equations

We prove Schauder estimates for a class of non-local elliptic operators with kernel $K(y)=a(y)/|y|^{d+\sigma}$ and either Dini or H\"older continuous data. Here $0 < \sigma < 2$ is a constant and $a$ is a bounded measurable function, which is not necessarily to be homogeneous, regular, or symmetric. As an application, we prove that the operators give isomorphisms between the Lipschitz--Zygmund spaces $\Lambda^{\alpha+\sigma}$ and $\Lambda^\alpha$ for any $\alpha>0$. Several local estimates and an extension to operators with kernels $K(x,y)$ are also discussed.