scholarly journals Boundedness and monotonicity of principal eigenvalues for boundary value problems with indefinite weight functions

2002 ◽  
Vol 30 (1) ◽  
pp. 25-29 ◽  
Author(s):  
G. A. Afrouzi
Keyword(s):  
Boundary Value Problem ◽  
Bounded Domain ◽  
Laplace Operator ◽  
Smooth Function ◽  
Smooth Boundary ◽  
Boundary Value ◽  
Weight Functions ◽  
Indefinite Weight ◽  
Principal Eigenvalues ◽  
Indefinite Weight Functions

We study the principal eigenvalues (i.e., eigenvalues corresponding to positive eigenfunctions) for the boundary value problem:−Δu(x)=λg(x)u(x),x∈D;(∂u/∂n)(x)+αu(x)=0,x∈∂D, whereΔis the standard Laplace operator,Dis a bounded domain with smooth boundary,g:D→ℝis a smooth function which changes sign onDandα∈ℝ. We discuss the relation betweenαand the principal eigenvalues.

scholarly journals On the continuity of principal eigenvalues for boundary value problems with indefinite weight function with respect to radius of balls inℝN

2002 ◽  
Vol 29 (5) ◽  
pp. 279-283
Author(s):  
Ghasem Alizadeh Afrouzi
Keyword(s):  
Boundary Value Problem ◽  
Weight Function ◽  
Boundary Value Problems ◽  
Smooth Function ◽  
Boundary Value ◽  
Continuous Functions ◽  
Indefinite Weight ◽  
Principal Eigenvalues ◽  
Indefinite Weight Function

We investigate the continuity of principal eigenvalues (i.e., eigenvalues corresponding to positive eigenfunctions) for the boundary value problem−Δu(x)=λg(x)u(x),x∈BR(0);u(x)=0,|x|=R, whereBR(0)is a ball inℝN, andgis a smooth function, and we show thatλ1+(R)andλ1−(R)are continuous functions ofR.

scholarly journals On the existence results for a class of singular elliptic system involving indefinite weight functions and asymptotically linear growth forcing term

2017 ◽  
Vol 37 (3) ◽  
pp. 67-74
Author(s):  
Ghasem A. Afrouzi ◽  
S. Shakeri ◽  
N. T. Chung
Keyword(s):  
Linear Growth ◽  
Smooth Boundary ◽  
Singular System ◽  
Existence Results ◽  
Weight Functions ◽  
Indefinite Weight ◽  
Asymptotically Linear ◽  
Existence Of Positive Solutions ◽  
Indefinite Weight Functions ◽  
Nondecreasing Functions

In this work, we study the existence of positive solutions to the singular system$$\left\{\begin{array}{ll}-\Delta_{p}u = \lambda a(x)f(v)-u^{-\alpha} & \textrm{ in }\Omega,\\-\Delta_{p}v = \lambda b(x)g(u)-v^{-\alpha} & \textrm{ in }\Omega,\\u = v= 0 & \textrm{ on }\partial \Omega,\end{array}\right.$$where $\lambda $ is positive parameter, $\Delta_{p}u=\textrm{div}(|\nabla u|^{p-2} \nabla u)$, $p>1$, $ \Omega \subset R^{n} $ some for $ n >1 $, is a bounded domain with smooth boundary $\partial \Omega $ , $ 0<\alpha< 1 $, and $f,g: [0,\infty] \to\R$ are continuous, nondecreasing functions which are asymptotically $ p $-linear at $\infty$. We prove the existence of a positive solution for a certain range of $\lambda$ using the method of sub-supersolutions.

scholarly journals On the estimates of eigenvalues of the boundary value problem with large parameter

2015 ◽  
Vol 63 (1) ◽  
pp. 101-113 ◽  
Author(s):  
Alexey V. Filinovskiy
Keyword(s):  
Eigenvalue Problem ◽  
Boundary Value Problem ◽  
Bounded Domain ◽  
Laplace Operator ◽  
Boundary Value ◽  
Dirichlet Condition ◽  
Large Parameter ◽  
Robin Condition ◽  
The Difference ◽  
The Laplace Operator

Abstract We consider the eigenvalue problem Δu + λu = 0 in Ω with Robin condition + αu = 0 on ∂Ω , where Ω ⊂ Rn , n ≥ 2 is a bounded domain and α is a real parameter. We obtain the estimates to the difference between λDk - λk(α) eigenvalue of the Laplace operator in with Dirichlet condition and the corresponding Robin eigenvalue for large positive values of for all k = 1,2,… We also show sharpness of these estimates in the power of α.

scholarly journals Three nontrivial solutions of boundary value problems for semilinear $\Delta_{\gamma}-$Laplace equation

2022 ◽  
Vol 40 ◽  
pp. 1-10
Author(s):  
Duong Trong Luyen ◽  
Le Thi Hong Hanh
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Bounded Domain ◽  
Morse Theory ◽  
Multiple Solutions ◽  
Smooth Boundary ◽  
Boundary Value ◽  
Nontrivial Solutions ◽  
Subcritical Growth ◽  
Subelliptic Operator

In this paper, we study the existence of multiple solutions for the boundary value problem\begin{equation}\Delta_{\gamma} u+f(x,u)=0 \quad \mbox{ in } \Omega, \quad \quad u=0 \quad \mbox{ on } \partial \Omega, \notag\end{equation}where $\Omega$ is a bounded domain with smooth boundary in $\mathbb{R}^N \ (N \ge 2)$ and $\Delta_{\gamma}$ is the subelliptic operator of the type $$\Delta_\gamma: =\sum\limits_{j=1}^{N}\partial_{x_j} \left(\gamma_j^2 \partial_{x_j} \right), \ \partial_{x_j}=\frac{\partial }{\partial x_{j}}, \gamma = (\gamma_1, \gamma_2, ..., \gamma_N), $$the nonlinearity $f(x , \xi)$ is subcritical growth and may be not satisfy the Ambrosetti-Rabinowitz (AR) condition. We establish the existence of three nontrivial solutions by using Morse theory.

scholarly journals Construction of a solution of a certain evolution equation II

1978 ◽  
Vol 71 ◽  
pp. 181-198 ◽  
Author(s):  
Akinobu Shimizu
Keyword(s):  
Brownian Motion ◽  
Evolution Equation ◽  
Boundary Value Problem ◽  
Bounded Domain ◽  
Smooth Boundary ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
One Dimensional ◽  
Initial Boundary Value

Let D be a bounded domain in Rd with smooth boundary ∂D. We denote by Bt, t ≥ 0, a one-dimensional Brownian motion. We shall consider the initial-boundary value problem

scholarly journals EXISTENCE OF INFINITELY MANY SOLUTIONS FOR SUBLINEAR ELLIPTIC PROBLEMS

2012 ◽  
Vol 54 (3) ◽  
pp. 535-545
Author(s):  
X. ZHONG ◽  
W. ZOU
Keyword(s):  
Boundary Value Problem ◽  
Bounded Domain ◽  
Dirichlet Boundary ◽  
Smooth Boundary ◽  
Boundary Value ◽  
Elliptic Problems ◽  
Infinitely Many Solutions ◽  
Dirichlet Boundary Value Problem ◽  
Image Position

AbstractWe study the following nonlinear Dirichlet boundary value problem: where Ω is a bounded domain in ℝN(N ≥ 2) with a smooth boundary ∂Ω and g ∈ C(Ω × ℝ) is a function satisfying $\displaystyle \underset{|t|\rightarrow 0}{\lim}\frac{g(x, t)}{t}= \infty$ for all x ∈ Ω. Under appropriate assumptions, we prove the existence of infinitely many solutions when g(x, t) is not odd in t.

UNIQUENESS FOR SINGULAR SEMILINEAR ELLIPTIC BOUNDARY VALUE PROBLEMS II

2015 ◽  
Vol 58 (2) ◽  
pp. 461-469 ◽  
Author(s):  
D. D. HAI ◽  
R. C. SMITH
Keyword(s):  
Boundary Value Problem ◽  
Bounded Domain ◽  
Smooth Boundary ◽  
Elliptic Boundary ◽  
Boundary Value ◽  
Positive Parameter ◽  
Elliptic Boundary Value ◽  
Semilinear Elliptic ◽  
Formula Group ◽  
Image Position

AbstractWe prove uniqueness of positive solutions for the boundary value problem \begin{equation*} \left\{ \begin{array}{l} -\Delta u=\lambda f(u)\text{ in }\Omega , \\ \ \ \ \ \ \ \ u=0\text{ on }\partial \Omega , \end{array} \right. \end{equation*} where Ω is a bounded domain in $\mathbb{R}$n with smooth boundary ∂ Ω, λ is a large positive parameter, f:(0,∞) → [0,∞) is nonincreasing for large t and is allowed to be singular at 0.

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