Nonlocal Elliptic Problems Research Articles

Multiplicity results for nonlocal critical elliptic problems

AbstractWe prove new multiplicity results for some nonlocal critical growth elliptic problems in bounded domains. More specifically, we show that the problems considered here have arbitrarily many solutions for all sufficiently large values of a certain parameter . In particular, the number of solutions goes to infinity as . We also give an explicit lower bound on in order to have a given number of solutions. This lower bound is in terms of a sequence of eigenvalues of the associated nonlocal elliptic operator. The proofs are based on an abstract critical point theorem.

Existence and uniqueness of positive solution to a new class of nonlocal elliptic problem with parameter dependency

In this paper, we prove that under weak assumptions on the reaction terms and diffusion coefficients, a positive solution exists for a one-dimensional case and a positive radial solution to a multidimensional case of a nonlocal elliptic problem. Additionally, we establish the uniqueness of the solution, with the fixed point theorem being the primary tool employed. Our results are new and generalize several existing results.

Lefschetz Formula for Nonlocal Elliptic Problems Associated with a Bundle

Lefschetz Formula for Nonlocal Elliptic Problems Associated with a Bundle

Fractional Musielak spaces: study of nonlocal elliptic problem with Choquard-logarithmic nonlinearity

In this study, we thoroughly examine a specific type of complex mathematical problems involving a fractional ψ κ , y ( ⋅ ) -Laplacian operator and a Choquard-logarithmic nonlinearity, combined with a real parameter, known as non-homogeneous elliptic problems. Our approach involves setting-specific conditions for the Choquard nonlinearities and the continuous function ψ κ , y . This allows us to successfully identify multiple solutions to these problems. Our analysis is conducted in the area of fractional Musielak spaces. We rely heavily on the mountain pass theorem and Ekeland's variational principle, as well as the Hardy–Littlewood–Sobolev inequality, which is essential for supporting the theoretical basis of our research.

Symmetrization results for general nonlocal linear elliptic and parabolic problems

We establish a Talenti-type symmetrization result in the form of mass concentration (i.e. integral comparison) for very general linear nonlocal elliptic problems, equipped with homogeneous Dirichlet boundary conditions.In this framework, the relevant concentration comparison for the classical fractional Laplacian can be reviewed as a special case of our main result, thus generalizing the previous results in [20].Finally, using an implicit time discretization techniques, similar results are obtained for the solutions of Cauchy-Dirichlet nonlocal linear parabolic problems.

Nonlinear nonlocal elliptic problems in ℝ3: existence results and qualitative properties

Nonlinear nonlocal elliptic problems in ℝ³: existence results and qualitative properties

Fundamental solutions and critical Lane-Emden exponents for nonlinear integral operators in cones

In this article we study the fundamental solutions or “α-harmonic functions” for some nonlinear positive homogeneous nonlocal elliptic problems in conical domains, such asF(u)=0inCω,u=0inRN∖Cω, where ω is a proper C2 domain in SN−1 for N≥2, Cω:={x:x≠0,|x|−1x∈ω} is the cone-like domain related to ω, and F is an extremal fully nonlinear integral operator. We prove the existence of two fundamental solutions that are homogeneous and do not change signs in the cone; one is bounded at the origin and the other at infinity.As an application, we use the fundamental solutions obtained to prove Liouville type theorems in cones for supersolutions of the Lane-Emden-Fowler equation in the formF(u)+up=0inCω,u=0inRN∖Cω. We also prove a generalized Hopf type lemma in domains with corners. Most of our results are new even when F is the fractional Laplacian operator.

Semilinear elliptic problems via the nonlinear Rayleigh quotient with two nonlocal nonlinearities

It is establish existence and multiplicity of solutions for nonlocal elliptic problems where the nonlinearity is driven by two convolutions terms. More specifically, we shall consider the following Choquard type problem: { − Δ u + V ( x ) u = μ ( I α 1 ∗ | u | q ) | u | q − 2 u − λ ( I α 2 ∗ | u | p ) | u | p − 2 u in R N , u ∈ H 1 ( R N ) , where p > q , λ , μ > 0 , α 1 ≤ α 2 ; α 1 , α 2 ∈ ( 0 , N ) , N ≥ 3 ; p ∈ ( 2 α 2 , 2 α 2 ∗ ) ; q ∈ ( 2 α 1 , 2 α 1 ∗ ) , 2 α j = ( N + α j ) / N and 2 α j ∗ = ( N + α j ) / ( N − 2 ) , j = 1 , 2 . Here we employ some variational arguments together with the Nehari method and the nonlinear Rayleigh quotient. The main feature in the present work is to find a sharp μ n > 0 and λ ∗ , λ ∗ > 0 such that our main problem admits at least two solutions for each μ > μ n where λ ∈ ( 0 , min ( λ ∗ , λ ∗ ) ) . The main difficulty here is to prove that the infimum associated to the energy functional restricted to the Nehari set is a weak solution for our main problem. This phenomenon occurs since the fibering maps for the associated energy functional have inflection points. Furthermore, we prove a nonexistence result for our main problem for each μ < μ n and λ > 0 .

On the existence of positive solutions of some nonlocal elliptic problems involving fractional p–Laplacian operator

Our aim in this paper is to analyze the existence of solutions to a nonlocal elliptic problem involving the fractional Laplacian operator. These operators are utilized to solve an equation defined within a bounded domain in . The operator is a fractional Laplacian , where with the condition and is a non-negative function almost everywhere with respect to the variable , and are real numbers. The article establishes two existence theorems for weak solutions using Tychonoff and Schauder fixed-point theorems. These theorems are formulated based on different hypotheses regarding the parameters and .

Kirchhoff type mixed local and nonlocal elliptic problems with concave–convex and Choquard nonlinearities

Kirchhoff type mixed local and nonlocal elliptic problems with concave–convex and Choquard nonlinearities

Existence of solution for a nonlocal elliptic problem with a Robin-type data

The aim of this work is to study the elliptic problem: 0 \ extrm{ in } \\Omega,\\\\ \\mathcal{N}_p^su+\\alpha(x)|u|^{p-2}u&=0 \ extrm{ in } \\mathbb{R}^N\\setminus\\Omega, \\end{align*} $$]]> ( − Δ p ) s u + | u | p − 2 u = μf ( x , u ) + λ u − γ in Ω , u > 0 in Ω , N p s u + α ( x ) | u | p − 2 u = 0 in R N ∖ Ω , under some conditions on the nonlinearity f. Here 2 ≤ p ≤ p s ∗ , 0 $ ]]> μ , λ > 0 , 0 < s , γ < 1 , α ∈ L ∞ ( R N ∖ Ω ) . We will prove the existence of at least one solution using the variational principle and the Mountain pass theorem.

Nonlocal critical growth elliptic problems with jumping nonlinearities

In this paper we study a nonlocal critical growth elliptic problem driven by the fractional Laplacian in the presence of a jumping nonlinearity. By using variational and topological methods and applying some new linking theorems recently proved by Perera and Sportelli in [19], we prove the existence of a nontrivial solution for the problem under consideration.The results we obtain here are the nonlocal counterparts of the ones obtained in [19] in the context of a local equation. Due to the nonlocal nature of our problem, some additional difficulties arise, and the arguments employed in the local setting need to be improved or reconceived. In fact, the proofs of our main theorems require some refined techniques and new regularity results for weak solutions of nonlocal problems that are of independent interest.We would like to point out that our results are specifically for a nonlocal problem with the fractional operator in integral form. However, we do not exclude the possibility that our results may have a counterpart for the spectral operator studied in [27]. Since nonlocal operators in integral form are being widely investigated in the current literature, especially in connection with geometric problems, we have restricted ourselves to elliptic equations driven by a fractional operator in integral form here.

Positive solutions, positive radial solutions and uniqueness results for some nonlocal elliptic problems

Positive solutions, positive radial solutions and uniqueness results for some nonlocal elliptic problems

On topological degree for pseudomonotone operators in fractional Orlicz-Sobolev spaces: study of positive solutions of non-local elliptic problems

On topological degree for pseudomonotone operators in fractional Orlicz-Sobolev spaces: study of positive solutions of non-local elliptic problems

Multiple solutions for critical nonlocal elliptic problems with magnetic field

In this paper, we consider the existence of multiple solutions of the following critical nonlocal elliptic equations with magnetic field: \begin{document}$\begin{align} \left\{\begin{aligned} (-i\nabla-A(x))^2u&amp; = \lambda |u|^{p-2}u+\left(\int_{\Omega}\frac{|u(y)|^{2^*_\alpha}}{|x-y|^{\alpha}}dy\right)|u|^{2^*_\alpha-2}u\quad {\rm in}\quad \Omega,\\ &amp;u = 0\quad \partial\Omega,\\ \end{aligned}\right. \end{align}\;\;\;\;(1)$ \end{document} where $ i $ is imaginary unit, $ N\geq4 $, $ 2^*_\alpha = \frac{2N-\alpha}{N-2} $ with $ 0&lt;\alpha&lt;4 $, $ \lambda&gt;0 $ and $ 2\leq p&lt;2^* = \frac{2N}{N-2} $. Suppose the magnetic vector potential $ A(x) = (A_1(x), A_2(x),..., A_N(x)) $ is real and local Hölder continuous, We show by the Ljusternik-Schnirelman theory that (1) has at least $ cat_\Omega(\Omega) $ nontrivial solutions for $ \lambda $ small.

Fractional Sobolev Spaces with Kernel Function on Compact Riemannian Manifolds

In this paper, a new class of Sobolev spaces with kernel function satisfying a Lévy-integrability-type condition on compact Riemannian manifolds is presented. We establish the properties of separability, reflexivity, and completeness. An embedding result is also proved. As an application, we prove the existence of solutions for a nonlocal elliptic problem involving the fractional p(·,·)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p(\\cdot , \\cdot )$$\\end{document}-Laplacian operator. As one of the main tools, topological degree theory is applied.

Comparison results for a nonlocal singular elliptic problem

We provide symmetrization results in the form of mass concentration comparisons for fractional singular elliptic equations in bounded domains, coupled with homogeneous external Dirichlet conditions. Two types of comparison results are presented, depending on the summability of the right-hand side of the equation. The maximum principle arguments employed in the core of the proofs of the main results offer a nonstandard, flexible alternative to the ones described in (Arch. Ration. Mech. Anal. 239 (2021) 1733–1770, Theorem 31). Some interesting consequences are L p regularity results and nonlocal energy estimates for solutions.

Simulating accurate and effective solutions of some nonlinear nonlocal two-point BVPs: Clique and QLM-clique matrix methods

This research work deals with two spectral matrix collocation algorithms based on (novel) clique functions to solve two classes of nonlinear nonlocal elliptic two-points boundary value problems (BVPs) arising in diverse physical models. The nonlinearity together with nonlocality makes the models so difficult to solve numerically. In both matrix methods by expanding the unknown solutions in terms of clique polynomials the nonlocality in the models is handled. In the first and direct technique, the clique coefficients are determined in an accurate way through solving an algebraic system of nonlinear equations. To get rid of the nonlinearity of the models and in order to gain efficacy, the quasilinearization method (QLM) is utilized in the second approach. In the latter algorithm named QLM-clique procedure, the first directly clique collocation method is applied to a family of linearized equations in an iterative manner. In particular, we show that the series expansion of clique polynomials is convergent exponentially in a weighted L2 norm. Numerous numerical simulations validate the performance of two matrix collocation schemes and demonstrate the accuracy as well as the gain in computational efficiency in terms of elapsed CPU time. The proposed matrix algorithms for computation are easy to implement and straightforward, and provide more accuracy compared to other available computational values in the literature.

Superlinear fractional elliptic problems via the nonlinear Rayleigh quotient with two parameters

AbstractIt is established existence of weak solutions for nonlocal elliptic problems driven by the fractional Laplacian where the nonlinearity is indefinite in sign. More specifically, we shall consider the following nonlocal elliptic problem: where , and . The potentials satisfy some extra assumptions. The main feature is to find sharp parameters and where the Nehari method can be applied. In order to do that, we employ the nonlinear Rayleigh quotient together a fine analysis on the fibering maps associated to the energy functional. It is important to mention also that for each parameters and , there exist degenerate points in the Nehari set that give serious difficulties. Furthermore, we consider nonlinearities that are superlinear at the origin and superlinear at infinity. In order to overcome these difficulties, we apply some estimates together with a carefully analysis on the fibering maps. Here, we also consider the asymptotic behavior of the weak solutions for our main problem when or . Furthermore, we consider a nonexistence result for our main problem under some appropriate conditions on the parameters and .

On Nonlocal Elliptic Problems of the Kirchhoff Type Involving the Hardy Potential and Critical Nonlinearity

In this article, we deal with the nonlocal elliptic problems of the Kirchhoff type involving the Hardy potential and critical nonlinearity on a bounded domain in R 3 . Under an appropriate condition on the nonhomogeneous term and using variational methods, we obtain two distinct solutions.