Nonlocal Elliptic Equations Research Articles

Multiplicity of solutions for a nonlocal nonhomogeneous elliptic equation with critical exponential growth

In this paper we are interested in the following nonlocal nonhomogeneous elliptic equation in R2,−Δu+V(x)u=(1|x|μ⁎F(u)|x|β)f(u)|x|β+εh(x)inR2, where V is a positive continuous potential, 0<μ<2, β≥0, 2β+μ≤2, ε is a small parameter and F(s) is the primitive function of f(s). Suppose that the nonlinearity f(s) is of critical exponential growth in the sense of Trudinger-Moser inequality, we prove the existence and multiplicity of solutions by variational methods.

On fractional regularity methods for a class of nonlocal problems

In the past years, the phenomenon of fractional regularity has been addressed for a large class of linear and/or quasilinear differential operators, mostly, in terms of certain Besov spaces. As it turned out, for equations governed by the p-Laplacian, in general, the regularity of solutions appears in terms of functional spaces with nonlinear order of smoothness. Moreover, despite its own interest, fractional regularity methods may be used as a tool for the investigation of some Partial Differential Equations which are not usually addressed in this manner. Thus, the purpose of the present paper is to exploit such methods in order to provide some results regarding existence and regularity of solutions to a class nonlocal elliptic equations which are linked to the p-Laplacian. This is done by means of explicit a priori estimates regarding Lebesgue and Nikolskii spaces, which are part of the present contribution. As a consequence, this approach allows a relaxation on some of the standard conditions employed in this class of problems.

The magnetization ripple: A nonlocal stochastic PDE perspective

The magnetization ripple is a microstructure formed by the magnetization in a thin-film ferromagnet. It is triggered by the random orientation of the grains in the poly-crystalline material. In an approximation of the micromagnetic model, which is sketched in this paper, this leads to a nonlocal (and strongly anisotropic) elliptic equation in two dimensions with white noise as a right hand side. However, like in singular Stochastic PDE, this right hand side is too rough for the non-linearity in the equation. In order to develop a small-date well-posedness theory, we take inspiration of the Da Prato–Debussche approach to singular SPDE. To this aim, we develop a Schauder theory for the non-standard symbol |k1|3+k22.

Multiple solutions for Hardy nonlocal fractional elliptic equations in $R^N$

Multiple solutions for Hardy nonlocal fractional elliptic equations in $R^N$

Existence and multiplicity of solutions for Hardy nonlocal fractional elliptic equations involving critical nonlinearities

In this paper, systems of fractional Laplacian equations are investigated, which involve critical homogeneous nonlinearities and Hardy-type terms as follows $$\begin{aligned} {\left\{ \begin{array}{ll} &{}(- \Delta )^s u - \gamma \frac{u}{|x|^{2 s}} = \frac{\lambda \mu }{\mu + \eta } \frac{|u|^{\mu - 2} |v|^{\eta } u}{|x|^{\alpha }}+ \frac{Q(x)}{2^*_s (\alpha )}\frac{ H_u(u,v)}{|x|^{\alpha }}, ~ in~ {\mathbb {R}}^N,\\ \\ &{} (- \Delta )^s v - \gamma \frac{v}{|x|^{2 s}} = \frac{\lambda \eta }{\mu + \eta } \frac{|u|^{\mu } |v|^{\eta - 2} v}{|x|^{\alpha }} + \frac{Q(x)}{2^*_s (\alpha )}\frac{ H_v(u,v)}{|x|^{\alpha }}, ~ in ~{\mathbb {R}}^N, \end{array}\right. } \end{aligned}$$ where $$0< s < 1$$ , $$0< \alpha< 2s <N$$ , $$0< \gamma < \gamma _H$$ with $$\begin{aligned} \gamma _H =4^s \frac{\Gamma ^2(\frac{N+2s}{4})}{\Gamma ^2(\frac{N-2s}{4})}, \;\;\; \text {for} \; i = 1,\ldots ,l \end{aligned}$$ being the fractional best Hardy constant on $${\mathbb {R}}^N$$ , $$2^*_s (\alpha ) = \frac{2 (N -\alpha )}{N - 2 s}$$ is critical Hardy–Sobolev exponent, $$\mu , \eta > 1$$ with $$\mu + \eta = 2^*_s(\alpha )$$ , Q is G-symmetric functions (G is a closed subgroup of O(N), see Sect. 2 for details) satisfying some appropriate conditions which will be specified later, $$\lambda $$ is real parameter, $$H_u$$ , $$H_v$$ are the partial derivatives of the 2-variable $$C^1$$ -functions H(u, v) and $$(-\Delta )^s$$ is the fractional Laplacian operator which (up to normalization factors) may be defined as $$\begin{aligned} (- \Delta )^s u (x) = - \frac{1}{2}\int _ {{\mathbb {R}}^{N} } \frac{u (x + y) + u (x - y) - 2 u (x) }{|x-y|^{N + 2 s}}{\text {d}}y. \end{aligned}$$ By variational methods and local compactness of Palais–Smale sequences, the extremals of the corresponding best Hardy–Sobolev constant are found and the existence of solutions to the system is established.

Nonlocal self-improving properties: a functional analytic approach

A functional analytic approach to obtaining self-improving properties of solutions to linear non-local elliptic equations is presented. It yields conceptually simple and very short proofs of some previous results due to Kuusi-Mingione-Sire and Bass-Ren. Its flexibility is demonstrated by new applications to non-autonomous parabolic equations with non-local elliptic part and questions related to maximal regularity

Nonlocal problems arising from the birth-jump processes

In this paper, we prove the existence and uniqueness of a positive solution for a nonlocal logistic equation arising from the birth-jump processes. For this, we establish a sub-super solution method for nonlocal elliptic equations, we perform a study of the eigenvalue problems associated with these equations and we apply these results to the nonlocal logistic equation.

Existence and concentration of solutions for singularly perturbed doubly nonlocal elliptic equations

We consider the existence and concentration of positive solutions to a singularly perturbed doubly nonlocal elliptic equation [Formula: see text] where [Formula: see text] is a parameter, [Formula: see text] are constants, [Formula: see text] and [Formula: see text] is an external potential, [Formula: see text] and [Formula: see text]. Under some suitable assumptions on [Formula: see text] and [Formula: see text], by using the penalization method, we prove that for [Formula: see text] small enough there exists a family of positive solutions which concentrate on the local minimum points of the potential [Formula: see text] as [Formula: see text].

Positive solutions of a nonlocal singular elliptic equation by means of a non-standard bifurcation theory

We consider a nonlocal elliptic equation arising in a prey–predator model whose nonlocal term is singular. We use the Leray–Schauder degree to prove the existence of an unbounded continuum of positive solutions emanating from the trivial solution. As application, we study nonlocal and singular elliptic equations of the type logistic and Holling–Tanner.

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.

Compactness of solutions to nonlocal elliptic equations

We show that all nonnegative solutions of the critical semilinear elliptic equation involving the regional fractional Laplacian are locally universally bounded. This strongly contrasts with the standard fractional Laplacian case. Secondly, we consider the fractional critical elliptic equations with nonnegative potentials. We prove compactness of solutions provided the potentials only have non-degenerate zeros. Corresponding to Schoen's Weyl tensor vanishing conjecture for the Yamabe equation on manifolds, we establish a Laplacian vanishing rate of the potentials at blow-up points of solutions.

Thin layer analysis of a non-local model for the double layer structure

For the structure of the thin electrical double layer (EDL) and the property related to the EDL capacitance, we analyze boundary layer solutions (corresponding to the electrostatic potential) of a non-local elliptic equation which is a steady-state Poisson–Nernst–Planck equation with a singular perturbation parameter related to the small Debye screening length. Theoretically, the boundary layer solutions describe that those ions exactly approach neutrality in the bulk, and the extra charges are accumulated near the charged surface. Hence, the non-neutral phenomenon merely occurs near the charged surface. To investigate such phenomena, we develop new analysis techniques to investigate thin boundary layer structures. A series of fine estimates combining the Pohožaev's identity, the inverse Hölder type estimates and some technical comparison arguments are developed in arbitrary bounded domains. Moreover, we focus on the physical domain being a ball with the simplest geometry and gain a clear picture on the effect of the curvature on the boundary layer solutions. The content involves three contributions. The first one focuses mainly on the boundary concentration phenomena. We show that the net charge density behaves exactly as Dirac delta measures concentrated at boundary points. The second one is devoted to pointwise descriptions with curvature effects for the thin boundary layer. An interesting outcome shows that the significant curvature effect merely occurs in the part of the boundary layer close to the boundary, and this part is extremely thinner than the whole boundary layer. The third contribution gives a connection to the EDL capacitance. We provide a theoretical way to support that the EDL has higher capacitance in a quite thin region near the charged surface, not in the whole EDL. In particular, for the cylindrical electrode, our result has a same analogous measurement as the specific capacitance of the well-known Helmholtz double layer.

Multiplicity of solutions for a class of nonlocal regional elliptic equations

In this article we are interested in the nonlinear Schrödinger equation with nonlocal regional diffusion:(0.1)(−Δ)ραu+u=f(x,u) in Rn,u∈Hα(Rn), where α∈(0,1) and (−Δ)ρα is a variational version of the regional Laplacian, whose range of scope is a ball with radius ρ(x)>0. By using the symmetric mountain pass theorem and the genus properties in critical point theory, we show that problem (0.1) has infinitely many solutions. Recent results in the literature are significantly improved.

Bubble-tower solutions to asymptotically critical nonlocal elliptic equations on conformal infinities

Assume that (X,g+) is an asymptotically hyperbolic manifold with conformal infinity (M,[hˆ]). We construct positive and sign-changing bubble-tower solutions to low-order perturbations of γ-Yamabe equations (0<γ<1) on (M,hˆ). These solutions exhibit isolated but non-simple blow-up in the sense of Schoen [30].

Sharp boundary behaviour of solutions to semilinear nonlocal elliptic equations

We investigate quantitative properties of nonnegative solutions $$u(x)\ge 0$$ to the semilinear diffusion equation $$\mathcal {L}u= f(u)$$ , posed in a bounded domain $$\Omega \subset \mathbb {R}^N$$ with appropriate homogeneous Dirichlet or outer boundary conditions. The operator $$\mathcal {L}$$ may belong to a quite general class of linear operators that include the standard Laplacian, the two most common definitions of the fractional Laplacian $$(-\Delta )^s$$ ( $$0<s<1$$ ) in a bounded domain with zero Dirichlet conditions, and a number of other nonlocal versions. The nonlinearity f is increasing and looks like a power function $$f(u)\sim u^p$$ , with $$p\le 1$$ . The aim of this paper is to show sharp quantitative boundary estimates based on a new iteration process. We also prove that, in the interior, solutions are Hölder continuous and even classical (when the operator allows for it). In addition, we get Hölder continuity up to the boundary. Particularly interesting is the behaviour of solution when the number $$\frac{2s}{1-p}$$ goes below the exponent $$\gamma \in (0,1]$$ corresponding to the Hölder regularity of the first eigenfunction $$\mathcal {L}\Phi _1=\lambda _1 \Phi _1$$ . Indeed a change of boundary regularity happens in the different regimes $$\frac{2s}{1-p} \gtreqqless \gamma $$ , and in particular a logarithmic correction appears in the “critical” case $$\frac{2s}{1-p} = \gamma $$ . For instance, in the case of the spectral fractional Laplacian, this surprising boundary behaviour appears in the range $$0<s\le (1-p)/2$$ .

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.

On Fourth-Order Elliptic Equations of Kirchhoff Type with Dependence on the Gradient and the Laplacian

We consider a nonlocal fourth-order elliptic equation of Kirchhoff type with dependence on the gradient and Laplacian Δ2u-a+b∫Ω∇u2dxΔu=fx,u,∇u,Δu, in Ω, u=0, Δu=0, on ∂Ω, where a, b are positive constants. We will show that there exists b⁎&gt;0 such that the problem has a nontrivial solution for 0&lt;b&lt;b⁎ through an iterative method based on the mountain pass lemma and truncation method developed by De Figueiredo et al., 2004.

Sign-changing solutions for non-local elliptic equations with asymptotically linear term

In this article, we study the existence of sign-changing solutions for a problem driven by a non-local integrodifferential operator with homogeneous Dirichlet boundary condition \begin{document}$\left\{ \begin{array}{ll}-\mathcal{L}_Ku = f(x,u) &\text{in}~Ω, \\u = 0 &\text{in}~\mathbb{R}^n\setminusΩ, \end{array} \right. \left( 1 \right)$ \end{document} where $Ω\subset\mathbb{R}^n(n≥2)$ is a bounded, smooth domain and $f(x, u)$ is asymptotically linear at infinity with respect to $u$. By introducing some new ideas and combining constraint variational method with the quantitative deformation lemma, we prove that there exists a sign-changing solution of problem (1).

Sign-changing solutions for non-local elliptic equations involving the fractional Laplacain

In this paper, we consider the existence of sign-changing solutions for fractional elliptic equations of the form \begin{equation*} \left\{\begin{array}{ll} (-\Delta)^s u=f(x,u) & \text{in}\ \Omega , \\ u=0 & \text{in}\ \mathbb R^N\setminus \Omega, \end{array} \right. \end{equation*} where $s\in(0,1)$ and $\Omega\subset \mathbb R^N$ is a bounded smooth domain. Since the non-local operator $(-\Delta)^s$ is involved in the equation, the variational functional of the equation has totally different properties from the local cases. By introducing some new ideas, we prove, via variational method and the method of invariant sets of descending flow, that the problem has a positive solution, a negative solution and a sign-changing solution under suitable conditions. Moreover, if $f(x,u)$ satisfies a monotonicity condition, we show that the problem has a least energy sign-changing solution with its energy is strictly larger than that of the ground state solution of Nehari type. We also obtain an unbounded sequence of sign-changing solutions if $f(x,u)$ is odd in $u$.

Existence in the Sense of Sequences of Stationary Solutions for Some Non-Fredholm Integro-Differential Equations

We establish the existence in the sense of sequences of stationary solutions for some reaction-diffusion type equations in appropriate H2 spaces. It is shown that, under reasonable technical conditions, the convergence in L1 of the integral kernels implies the existence and convergence in H2 of solutions. The nonlocal elliptic equations involve second order differential operators with and without the Fredholm property. Bibliography: 21 titles.