EIGENVALUE PROBLEMS ASSOCIATED WITH NONHOMOGENEOUS DIFFERENTIAL OPERATORS, IN ORLICZ–SOBOLEV SPACES

2008 ◽  
Vol 06 (01) ◽  
pp. 83-98 ◽  
Author(s):  
MIHAI MIHĂILESCU ◽  
VICENŢIU RĂDULESCU
Keyword(s):  
Continuous Function ◽  
Boundary Value Problem ◽  
Real Number ◽  
Bounded Domain ◽  
Sobolev Spaces ◽  
Positive Real Number ◽  
Differential Operators ◽  
Eigenvalue Problems ◽  
Smooth Boundary ◽  
Positive Real

We study the boundary value problem - div ((a1(|∇ u|) + a2(|∇ u|))∇ u) = λ|u|q(x)-2u in Ω, u = 0 on ∂Ω, where Ω is a bounded domain in ℝN (N ≥ 3) with smooth boundary, λ is a positive real number, q is a continuous function and a1, a2 are two mappings such that a1(|t|)t, a2(|t|)t are increasing homeomorphisms from ℝ to ℝ. We establish the existence of two positive constants λ0 and λ1 with λ0 ≤ λ1 such that any λ ∈ [λ1, ∞) is an eigenvalue, while any λ ∈ (0, λ0) is not an eigenvalue of the above problem.

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.

scholarly journals A continuous spectrum for nonhomogeneous differential operators in Orlicz-Sobolev spaces

2009 ◽  
Vol 104 (1) ◽  
pp. 132 ◽  
Author(s):  
Mihai Mihailescu ◽  
Vicentiu Radulescu
Keyword(s):  
Continuous Function ◽  
Eigenvalue Problem ◽  
Continuous Spectrum ◽  
Sobolev Spaces ◽  
Differential Operators ◽  
Smooth Boundary ◽  
Nonlinear Eigenvalue Problem ◽  
Sufficient Conditions ◽  
Open Set ◽  
Quasilinear Problem

We study the nonlinear eigenvalue problem $-(\mathrm{div} (a(|\nabla u|)\nabla u)=\lambda|u|^{q(x)-2}u$ in $\Omega$, $u=0$ on $\partial\Omega$, where $\Omega$ is a bounded open set in ${\mathsf R}^N$ with smooth boundary, $q$ is a continuous function, and $a$ is a nonhomogeneous potential. We establish sufficient conditions on $a$ and $q$ such that the above nonhomogeneous quasilinear problem has continuous families of eigenvalues. The proofs rely on elementary variational arguments. The abstract results of this paper are illustrated by the cases $a(t)=t^{p-2}\log (1+t^r)$ and $a(t)= t^{p-2} [\log (1+t)]^{-1}$.

The convergence of nonnegative solutions for the family of problems − Δpu = λeu as p →∞

2018 ◽  
Vol 24 (2) ◽  
pp. 569-578
Author(s):  
Mihai Mihăilescu ◽  
Denisa Stancu−Dumitru ◽  
Csaba Varga
Keyword(s):  
Boundary Condition ◽  
Real Number ◽  
Positive Real Number ◽  
Dirichlet Boundary ◽  
Smooth Boundary ◽  
Nonnegative Solution ◽  
Homogeneous Dirichlet Boundary Condition ◽  
Positive Real ◽  
Nonnegative Solutions ◽  
The Family

Let Ω ⊂ ℝN (N ≥ 2) be a bounded domain with smooth boundary. We show the existence of a positive real number λ* such that for each λ ∈ (0, λ*) and each real number p > N the equation −Δp u = λeu in Ω subject to the homogeneous Dirichlet boundary condition possesses a nonnegative solution up. Next, we analyze the asymptotic behavior of up as p → ∞ and we show that it converges uniformly to the distance function to the boundary of the domain.

scholarly journals Existence results for nonlinear problems with $\varphi$- Laplacian operators and nonlocal boundary conditions

2022 ◽  
Vol 40 ◽  
pp. 1-10
Author(s):  
Dionicio Pastor Dallos Santos
Keyword(s):  
Continuous Function ◽  
Real Number ◽  
Positive Real Number ◽  
Nonlocal Boundary ◽  
Degree Theory ◽  
Nonlinear Problems ◽  
Nonlocal Boundary Conditions ◽  
Existence Results ◽  
Positive Real ◽  
Laplacian Operators

Using Leray-Schauder degree theory we study the existence of at least one solution for the boundary value problem of the type\[\left\{\begin{array}{lll}(\varphi(u' ))' = f(t,u,u') & & \\u'(0)=u(0), \ u'(T)= bu'(0), & & \quad \quad \end{array}\right.\] where $\varphi: \mathbb{R}\rightarrow \mathbb{R}$ is a homeomorphism such that $\varphi(0)=0$, $f:\left[0, T\right]\times \mathbb{R} \times \mathbb{R}\rightarrow \mathbb{R} $ is a continuous function, $T$ a positive real number, and $b$ some non zero real number.

Minimization problems for inhomogeneous Rayleigh quotients

2018 ◽  
Vol 20 (07) ◽  
pp. 1750074
Author(s):  
Marian Bocea ◽  
Mihai Mihăilescu
Keyword(s):  
Real Number ◽  
Distance Function ◽  
Minimization Problem ◽  
Positive Real Number ◽  
Smooth Boundary ◽  
Positive Real ◽  
Minimization Problems ◽  
Bounded Convex Domain ◽  
Principal Frequency ◽  
Bounded Convex

In this paper, the minimization problem [Formula: see text] where [Formula: see text] is studied when [Formula: see text] ([Formula: see text]) is an open, bounded, convex domain with smooth boundary and [Formula: see text]. We show that [Formula: see text] is either zero, when the maximum of the distance function to the boundary of [Formula: see text] is greater than [Formula: see text], or it is a positive real number, when the maximum of the distance function to the boundary of [Formula: see text] belongs to the interval [Formula: see text]. In the latter case, we provide estimates for [Formula: see text] and show that for [Formula: see text] sufficiently large [Formula: see text] coincides with the principal frequency of the [Formula: see text]-Laplacian in [Formula: see text]. Some particular cases and related problems are also discussed.

scholarly journals Problems with Mixed Boundary Conditions in Banach Spaces

2017 ◽  
Vol 2017 ◽  
pp. 1-8 ◽  
Author(s):  
Dionicio Pastor Dallos Santos
Keyword(s):  
Banach Space ◽  
Continuous Function ◽  
Boundary Value Problem ◽  
Positive Real Number ◽  
Real Banach Space ◽  
Mixed Boundary ◽  
Lipschitz Constant ◽  
Mixed Boundary Conditions ◽  
Positive Real ◽  
Condensing Maps

Using Leray-Schauder degree or degree for α-condensing maps we obtain the existence of at least one solution for the boundary value problem of the following type: φu′′=ft,u,u′,  u(T)=0=u′(0), where φ:X→X is a homeomorphism with reverse Lipschitz constant such that φ(0)=0, f:0,T×X×X→X is a continuous function, T is a positive real number, and X is a real Banach space.

scholarly journals Finding of principle square root of a real number by using interpolation method

2018 ◽  
Vol 7 (1) ◽  
pp. 77-83
Author(s):  
Rajendra Prasad Regmi
Keyword(s):  
Real Number ◽  
Positive Real Number ◽  
Interpolation Method ◽  
Square Root ◽  
Real Numbers ◽  
Positive Real ◽  
Unknown Quantity ◽  
Square Roots

There are various methods of finding the square roots of positive real number. This paper deals with finding the principle square root of positive real numbers by using Lagrange’s and Newton’s interpolation method. The interpolation method is the process of finding the values of unknown quantity (y) between two known quantities.

Some remarks on the solvability of non-local elliptic problems with the Hardy potential

2014 ◽  
Vol 16 (04) ◽  
pp. 1350046 ◽  
Author(s):  
B. Barrios ◽  
M. Medina ◽  
I. Peral
Keyword(s):  
Real Number ◽  
Sobolev Inequality ◽  
Positive Real Number ◽  
Fractional Power ◽  
Existence Of Solution ◽  
Sharp Constant ◽  
Hardy Potential ◽  
Positive Real ◽  
Existence And Multiplicity ◽  
Non Local

The aim of this paper is to study the solvability of the following problem, [Formula: see text] where (-Δ)s, with s ∈ (0, 1), is a fractional power of the positive operator -Δ, Ω ⊂ ℝN, N > 2s, is a Lipschitz bounded domain such that 0 ∈ Ω, μ is a positive real number, λ < ΛN,s, the sharp constant of the Hardy–Sobolev inequality, 0 < q < 1 and [Formula: see text], with αλ a parameter depending on λ and satisfying [Formula: see text]. We will discuss the existence and multiplicity of solutions depending on the value of p, proving in particular that p(λ, s) is the threshold for the existence of solution to problem (Pμ).

Ultradiversities and their spherical completeness

2020 ◽  
Vol 26 (2) ◽  
pp. 231-240
Author(s):  
Gholamreza H. Mehrabani ◽  
Kourosh Nourouzi
Keyword(s):  
Real Number ◽  
Finite Subset ◽  
Positive Real Number ◽  
Metric Spaces ◽  
Nonexpansive Mappings ◽  
Ultrametric Spaces ◽  
Positive Real

AbstractDiversities are a generalization of metric spaces which associate a positive real number to every finite subset of the space. In this paper, we introduce ultradiversities which are themselves simultaneously diversities and a sort of generalization of ultrametric spaces. We also give the notion of spherical completeness for ultradiversities based on the balls defined in such spaces. In particular, with the help of nonexpansive mappings defined between ultradiversities, we show that an ultradiversity is spherically complete if and only if it is injective.

An eigenvalue problem for generalized Laplacian in Orlicz—Sobolev spaces

1999 ◽  
Vol 129 (1) ◽  
pp. 153-163 ◽  
Author(s):  
Vesa Mustonen ◽  
Matti Tienari
Keyword(s):  
Weak Solution ◽  
Continuous Function ◽  
Eigenvalue Problem ◽  
Bounded Domain ◽  
Sobolev Spaces ◽  
Minimization Problem ◽  
Lagrange Equation ◽  
Image Position ◽  
Generalized Laplacian

Let m: [ 0, ∞) → [ 0, ∞) be an increasing continuous function with m(t) = 0 if and only if t = 0, m(t) → ∞ as t → ∞ and Ω C ℝN a bounded domain. In this note we show that for every r > 0 there exists a function ur solving the minimization problemwhere Moreover, the function ur is a weak solution to the corresponding Euler–Lagrange equationfor some λ > 0. We emphasize that no Δ2-condition is needed for M or M; so the associated functionals are not continuously differentiable, in general.

Sign in / Sign up

Export Citation Format

Share Document