On an eigenvalue problem for an anisotropic elliptic equation

2019 ◽  
Author(s):  
Said Taarabti ◽  
Zakaria El Allali ◽  
Khalil Ben Haddouch
Keyword(s):  
Elliptic Equation ◽  
Eigenvalue Problem ◽  
Anisotropic Elliptic Equation

scholarly journals Some remarks on an eigenvalue problem for an anisotropic elliptic equation with indefinite weight

Filomat ◽  
2019 ◽  
Vol 33 (16) ◽  
pp. 5061-5075
Author(s):  
Nguyen Chung
Keyword(s):  
Differential Operator ◽  
Elliptic Equation ◽  
Eigenvalue Problem ◽  
Variational Methods ◽  
Growth Rates ◽  
Continuous Family ◽  
Variable Exponents ◽  
Indefinite Weight ◽  
Partial Derivatives ◽  
Anisotropic Elliptic Equation

In this paper, we consider an eigenvalue problem for an anisotropic elliptic equation with indefinite weight, in which the differential operator involves partial derivatives with different variable exponents. Under some suitable conditions on the growth rates of the anisotropic coefficients involved in the problem, we prove some results on the existence and non-existence of a continuous family of eigenvalues by using variational methods.

scholarly journals ON AN EIGENVALUE PROBLEM FOR AN ANISOTROPIC ELLIPTIC EQUATION INVOLVING VARIABLE EXPONENTS

2010 ◽  
Vol 52 (3) ◽  
pp. 517-527 ◽  
Author(s):  
MIHAI MIHĂILESCU ◽  
GHEORGHE MOROŞANU
Keyword(s):  
Elliptic Equation ◽  
Eigenvalue Problem ◽  
Real Number ◽  
Bounded Domain ◽  
Positive Real Number ◽  
Smooth Boundary ◽  
Continuous Functions ◽  
Variable Exponents ◽  
Positive Real ◽  
Anisotropic Elliptic Equation

AbstractWe study the eigenvalue problem $\(-\sum_{i=1}^N\di\partial_{x_i}(|\di\partial_{x_i}u |^{p_i(x)-2}\di\partial_{x_i}u)$ = λ|u|q(x)−2u in Ω, u = 0 on ∂Ω, where Ω is a bounded domain in ℝN with smooth boundary ∂Ω, λ is a positive real number, and p1,⋅ ⋅ ⋅, pN, q are continuous functions satisfying the following conditions: 2 ≤ pi(x) < N, 1 < q(x) for all x ∈ Ω, i ∈ {1,. . .,N}; there exist j, k ∈ {1,. . .,N}, j ≠ k, such that pj ≡ q in Ω, q is independent of xj and maxΩq < minΩpk. The main result of this paper establishes the existence of two positive constants λ0 and λ1 with λ0 ≤ λ1 such that every λ ∈(λ1, ∞) is an eigenvalue, while no λ ∈ (0, λ0) can be an eigenvalue of the above problem.

Positivity results for a nonlocal elliptic equation

1998 ◽  
Vol 128 (4) ◽  
pp. 697-715 ◽  
Author(s):  
Pedro Freitas ◽  
Guido Sweers
Keyword(s):  
Elliptic Equation ◽  
Eigenvalue Problem ◽  
Parabolic Equations ◽  
Elliptic Systems ◽  
Stationary Solutions ◽  
Second Order ◽  
Real Eigenvalue ◽  
Nonlocal Operator ◽  
Nonlocal Parabolic Equations ◽  
Positive Eigenfunction

In this paper we consider a second-order linear nonlocal elliptic operator on a bounded domain in ℝn (n ≧ 3), and give conditions which ensure that this operator has a positive inverse. This generalises results of Allegretto and Barabanova, where the kernel of the nonlocal operator was taken to be separable. In particular, our results apply to the case where this kernel is the Green's function associated with second-order uniformly elliptic operators, and thus include the case of some linear elliptic systems. We give several other examples. For a specific case which appears when studying the linearisation of nonlocal parabolic equations around stationary solutions, we also consider the associated eigenvalue problem and give conditions which ensure the existence of a positive eigenfunction associated with the smallest real eigenvalue.

Positive solutions of Kirchhoff-type non-local elliptic equation: a bifurcation approach

2017 ◽  
Vol 147 (4) ◽  
pp. 875-894 ◽  
Author(s):  
Zhanping Liang ◽  
Fuyi Li ◽  
Junping Shi
Keyword(s):  
Elliptic Equation ◽  
Eigenvalue Problem ◽  
Positive Solutions ◽  
Nonlinear Eigenvalue Problem ◽  
Global Bifurcation ◽  
Integral Term ◽  
Kirchhoff Type ◽  
Nonlinear Elliptic ◽  
Linear Eigenvalue Problem ◽  
Non Local

Positive solutions of a Kirchhoff-type nonlinear elliptic equation with a non-local integral term on a bounded domain in ℝN, N ⩾ 1, are studied by using bifurcation theory. The parameter regions of existence, non-existence and uniqueness of positive solutions are characterized by the eigenvalues of a linear eigenvalue problem and a nonlinear eigenvalue problem. Local and global bifurcation diagrams of positive solutions for various parameter regions are obtained.

scholarly journals Existence and multiplicity of solutions for anisotropic elliptic equation

2022 ◽  
Vol 40 ◽  
pp. 1-12
Author(s):  
El Amrouss Abdelrachid ◽  
Ali El Mahraoui
Keyword(s):  
Elliptic Equation ◽  
Variational Method ◽  
Nonlinear Problem ◽  
Multiplicity Of Solutions ◽  
Anisotropic Elliptic Equation ◽  
Existence And Multiplicity

In this article we study the nonlinear problem $$\left\{ \begin{array}{lr} -\sum_{i=1}^{N}\partial_{x_{i}}a_{i}(x,\partial_{x_{i}}u)+ b(x)~|u|^{P_{+}^{+}-2}u =\lambda f(x,u) \quad in \quad \Omega\\ u=0 \qquad on \qquad \partial\Omega \end{array} \right.$$ Using the variational method, under appropriate assumptions on $f$, we obtain a result on existence and multiplicity of solutions.

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