Solutions For Nonlocal Elliptic Problems Research Articles

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 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

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 .

Multiple solutions for nonlocal elliptic problems driven by $ p(x) $-biharmonic operator

<abstract> In this article, we study the existence of at least three distinct weak solutions for nonlocal elliptic problems involving $ p(x) $-biharmonic operator. The results are obtained by means of variational methods. We also provide an example in order to illustrate our main abstract results. We extend and improve some recent results. </abstract>

Existence and multiplicity of solutions for Nonlocal elliptic problem involving p−biharmonic operator and concave-convex nonlinearities

Existence and multiplicity of solutions for Nonlocal elliptic problem involving p−biharmonic operator and concave-convex nonlinearities

Multiple positive solutions for nonlocal elliptic problems involving the Hardy potential and concave–convex nonlinearities

In this paper, we study the existence and multiplicity of solutions for the following fractional problem involving the Hardy potential and concave–convex nonlinearities: [Formula: see text] where [Formula: see text] is a smooth bounded domain in [Formula: see text] containing [Formula: see text] in its interior, and [Formula: see text] with [Formula: see text] which may change sign in [Formula: see text]. We use the variational methods and the Nehari manifold decomposition to prove that this problem has at least two positive solutions for [Formula: see text] sufficiently small. The variational approach requires that [Formula: see text] [Formula: see text] [Formula: see text], and [Formula: see text], the latter being the best fractional Hardy constant on [Formula: see text].

Bubbling solutions for nonlocal elliptic problems

We investigate bubbling solutions for the nonlocal equation \[ A_\Omega^s u =u^p,\ u >0 \quad \mbox{in } \Omega, \] under homogeneous Dirichlet conditions, where $\Omega$ is a bounded and smooth domain. The operator $A_\Omega^s$ stands for two types of nonlocal operators that we treat in a unified way: either the spectral fractional Laplacian or the restricted fractional Laplacian. In both cases $s \in (0,1)$ and the Dirichlet conditions are different: for the spectral fractional Laplacian, we prescribe $u=0$ on $\partial \Omega$ and for the restricted fractional Laplacian, we prescribe $u=0$ on $\mathbb R^n \setminus \Omega$. We construct solutions when the exponent $p = (n+2s)/(n-2s) \pm \varepsilon$ is close to the critical one, concentrating as $\varepsilon \to 0$ near critical points of a reduced function involving the Green and Robin functions of the domain

Non-trivial solutions for nonlocal elliptic problems of Kirchhoff-type

Abstract Existence results of positive solutions for a nonlocal elliptic problem of Kirchhoff-type are established. The approach is based on variational methods.

Sequences of Weak Solutions for Non-Local Elliptic Problems with Dirichlet Boundary Condition

AbstractIn this paper the existence of infinitely many solutions for a class of Kirchhoff-type problems involving the p-Laplacian, with p &gt; 1, is established. By using variational methods, we determine unbounded real intervals of parameters such that the problems treated admit either an unbounded sequence of weak solutions, provided that the nonlinearity has a suitable behaviour at ∞, or a pairwise distinct sequence of weak solutions that strongly converges to 0 if a similar behaviour occurs at 0. Some comparisons with several results in the literature are pointed out. The last part of the work is devoted to the autonomous elliptic Dirichlet problem.

Asymptotic formulas for solutions of nonlocal elliptic problems

We consider nonlocal elliptic problems in plane domains and obtain asymptotic formulas for solutions in weighted spaces near junction points.

On the index instability for some nonlocal elliptic problems

The index of unbounded operators defined on generalized solutions of nonlocal elliptic problems in plane bounded domains is investigated. It is known that nonlocal terms with smooth coefficients having zero of a certain order at the conjugation points do not affect the index of the unbounded operator. In this paper, we construct examples showing that the index may change under nonlocal perturbations with coefficients not vanishing at the points of conjugation of boundary-value conditions.

The consistency conditions and the smoothness of generalized solutions of nonlocal elliptic problems

We study smoothness of generalized solutions of nonlocal elliptic problems in plane bounded domains with piecewise smooth boundary. The case where the support of nonlocal terms can intersect the boundary is considered. We find conditions that are necessary and sufficient for any generalized solution to possess an appropriate smoothness (in terms of Sobolev spaces). Both homogeneous and nonhomogeneous nonlocal boundary-value conditions are studied.

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 W 2 m (G) are studied. The Fredholm property of the unbounded operator (corresponding to the elliptic equation) acting on L 2(G), and defined for functions from the space W 2 m (G) that satisfy homogeneous nonlocal conditions, is established.

Asymptotics of Solutions for Nonlocal Elliptic Problems in Plane Angles

In the paper we investigate asymptotics of solutions for nonlocal elliptic problems in plane angles and in ℝ2 \ {0}. These problems arise in studying nonlocal problems in bounded domains in the case where support of nonlocal terms intersects with a boundary of a domain. We obtain explicit formulas for the asymptotic coefficients in terms of eigenvectors and associated vectors of both adjoint nonlocal operators acting in spaces of distributions and formally adjoint (with respect to the Green formula) nonlocal transmission problems.

Some properties of solutions of nonlocal elliptic problems in the space of generalized functions

Some properties of solutions of nonlocal elliptic problems in the space of generalized functions