An intermediate local–nonlocal eigenvalue elliptic problem

2020 ◽  
pp. 2050076
Author(s):  
Manuel Delgado ◽  
Joao R. Santos ◽  
Antonio Suárez
Keyword(s):  
Population Dynamics ◽  
Eigenvalue Problem ◽  
Elliptic Problem ◽  
Nonlocal Diffusion ◽  
Unknown Variable ◽  
Elliptic Eigenvalue Problem

This paper deals with a nonlocal diffusion elliptic eigenvalue problem. Specifically, the diffusion of the unknown variable at a point of the domain depends on its value in a neighborhood of the point. We apply bifurcation arguments and appropriate approximation to obtain our results. Some applications to the population dynamics will be given.

On the maximum of solutions for a semilinear elliptic problem

1988 ◽  
Vol 108 (3-4) ◽  
pp. 357-370 ◽  
Author(s):  
Guido Sweers
Keyword(s):  
Eigenvalue Problem ◽  
Elliptic Problem ◽  
Dimensional Case ◽  
Bounded Domains ◽  
One Dimensional ◽  
Elliptic Eigenvalue Problem ◽  
Semilinear Elliptic Problem ◽  
Semilinear Elliptic ◽  
Right Hand ◽  
The One

SynopsisIn this paper we study some properties of a semilinear elliptic eigenvalue problem with nondefinite right-hand side. In the first part we show that every solution will have its maximum in some specified interval J. If the domain is inside a cone in ℝN with N > 1, then J is strictly smaller than in the one-dimensional case. In the second part we show, for bounded domains, that if the maximum is inside some subinterval of J, then for any eigenvalue there will be at most one solution.

Asymptotic analysis of the eigenvalues of an elliptic problem in an anisotropic thin multidomain

2011 ◽  
Vol 141 (4) ◽  
pp. 739-754 ◽  
Author(s):  
Antonio Gaudiello ◽  
Ali Sili
Keyword(s):  
Eigenvalue Problem ◽  
Asymptotic Analysis ◽  
Elliptic Problem ◽  
Anisotropic Material ◽  
Elliptic Eigenvalue Problem

We study, via an asymptotic analysis, an elliptic eigenvalue problem in a 1D–1D multidomain and in a 1D–2D multidomain filled with anisotropic material. The corresponding isotropic cases were considered in a previous work by Gaudiello and Sili.

scholarly journals Remarks on a nonlinear nonlocal operator in Orlicz spaces

2019 ◽  
Vol 9 (1) ◽  
pp. 305-326 ◽  
Author(s):  
Ernesto Correa ◽  
Arturo de Pablo
Keyword(s):  
Boundary Condition ◽  
Eigenvalue Problem ◽  
Bounded Domain ◽  
Elliptic Problem ◽  
Sobolev Inequality ◽  
Orlicz Spaces ◽  
Integral Operators ◽  
Laplacian Operator ◽  
Nonlocal Operator ◽  
Generalized Eigenvalue

Abstract We study integral operators $\mathcal{L}u\left( \chi \right)=\int{_{_{\mathbb{R}}\mathbb{N}}\psi \left( u\left( x \right)-u\left( y \right) \right)J\left( x-y \right)dy}$of the type of the fractional p-Laplacian operator, and the properties of the corresponding Orlicz and Sobolev-Orlicz spaces. In particular we show a Poincaré inequality and a Sobolev inequality, depending on the singularity at the origin of the kernel J considered, which may be very weak. Both inequalities lead to compact inclusions. We then use those properties to study the associated elliptic problem $\mathcal{L}u=f$in a bounded domain $\Omega ,$and boundary condition u ≡ 0 on ${{\Omega }^{c}};$both cases f = f(x) and f = f(u) are considred, including the generalized eigenvalue problem $f\left( u \right)=\lambda \psi \left( u \right).$

scholarly journals Infinitely many solutions for non-local problems with broken symmetry

2018 ◽  
Vol 7 (3) ◽  
pp. 353-364
Author(s):  
Rossella Bartolo ◽  
Pablo L. De Nápoli ◽  
Addolorata Salvatore
Keyword(s):  
Eigenvalue Problem ◽  
Elliptic Problem ◽  
Existence Of Solutions ◽  
Broken Symmetry ◽  
Laplacian Operator ◽  
Infinitely Many Solutions ◽  
Lipschitz Boundary ◽  
Local Problems ◽  
Non Local ◽  
Fractional Laplace Operator

AbstractThe aim of this paper is to investigate the existence of solutions of the non-local elliptic problem\left\{\begin{aligned} &\displaystyle(-\Delta)^{s}u\ =\lvert u\rvert^{p-2}u+h(% x)&&\displaystyle\text{in }\Omega,\\ &\displaystyle{u=0}&&\displaystyle\text{on }\mathbb{R}^{n}\setminus\Omega,\end% {aligned}\right.where {s\in(0,1)}, {n>2s}, Ω is an open bounded domain of {\mathbb{R}^{n}} with Lipschitz boundary {\partial\Omega}, {(-\Delta)^{s}} is the non-local Laplacian operator, {2<p<2_{s}^{\ast}} and {h\in L^{2}(\Omega)}. This problem requires the study of the eigenvalue problem related to the fractional Laplace operator, with or without potential.

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