scholarly journals A Class of Degenerate Nonlinear Elliptic Equations in Weighted Sobolev Space

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Rasmita Kar
Keyword(s):  
Weak Solution ◽  
Sobolev Space ◽  
Boundary Value Problem ◽  
Elliptic Equations ◽  
Dirichlet Boundary ◽  
Weighted Sobolev Space ◽  
Nonlinear Elliptic Equations ◽  
Dirichlet Boundary Value Problem ◽  
Nonlinear Elliptic ◽  
Bounded Nonlinearity

We prove the existence of a weak solution for the degenerate nonlinear elliptic Dirichlet boundary-value problem Lu-μug1+hu,∇ug2=f in Ω, u=0 on ∂Ω, in a suitable weighted Sobolev space, where Ω⊂ℝn is a bounded domain and h is a continuous bounded nonlinearity.

A uniqueness principle for up ≤ (−Δ)α 2u in the Euclidean space

2016 ◽  
Vol 18 (06) ◽  
pp. 1650019 ◽  
Author(s):  
Y. Wang ◽  
J. Xiao
Keyword(s):  
Weak Solution ◽  
Euclidean Space ◽  
Fractional Order ◽  
Elliptic Equations ◽  
Differential Inequality ◽  
Nonlinear Elliptic Equations ◽  
Local Behavior ◽  
Nonlinear Elliptic ◽  
Global And Local ◽  
Appl Math

This paper establishes such a uniqueness principle that under [Formula: see text] the fractional order differential inequality [Formula: see text] has the property that if [Formula: see text] then a non-negative weak solution to [Formula: see text] is unique, and if [Formula: see text] then the uniqueness of a non-negative weak solution to [Formula: see text] occurs when and only when [Formula: see text], thereby innovatively generalizing Gidas–Spruck’s result for [Formula: see text] in [Formula: see text] discovered in [B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math. 34 (1981) 525–598].

scholarly journals Regularity of Extremal Solutions to Nonlinear Elliptic Equations with Quadratic Convection and General Reaction

2020 ◽  
Vol 17 (6) ◽  
Author(s):  
Asadollah Aghajani ◽  
Fatemeh Mottaghi ◽  
Vicenţiu D. Rădulescu
Keyword(s):  
Elliptic Equations ◽  
Nonlinear Elliptic Equation ◽  
Dirichlet Boundary ◽  
Nonlinear Elliptic Equations ◽  
Extremal Solution ◽  
General Reaction ◽  
Smooth Bounded Domain ◽  
Nonlinear Elliptic ◽  
Regularity Results ◽  
Increasing Function

AbstractWe consider the nonlinear elliptic equation with quadratic convection $$ -\Delta u + g(u) |\nabla u|^2=\lambda f(u) $$ - Δ u + g ( u ) | ∇ u | 2 = λ f ( u ) in a smooth bounded domain $$ \Omega \subset {\mathbb {R}}^N $$ Ω ⊂ R N ($$ N \ge 3$$ N ≥ 3 ) with zero Dirichlet boundary condition. Here, $$ \lambda $$ λ is a positive parameter, $$ f:[0, \infty ):(0\infty ) $$ f : [ 0 , ∞ ) : ( 0 ∞ ) is a strictly increasing function of class $$C^1$$ C 1 , and g is a continuous positive decreasing function in $$ (0, \infty ) $$ ( 0 , ∞ ) and integrable in a neighborhood of zero. Under natural hypotheses on the nonlinearities f and g, we provide some new regularity results for the extremal solution $$u^*$$ u ∗ . A feature of this paper is that our main contributions require neither the convexity (even at infinity) of the function $$ h(t)=f(t)e^{-\int _0^t g(s)ds}$$ h ( t ) = f ( t ) e - ∫ 0 t g ( s ) d s , nor that the functions $$ gh/h'$$ g h / h ′ or $$ h'' h/h'^2$$ h ′ ′ h / h ′ 2 admit a limit at infinity.

Non-existence of solutions for some nonlinear elliptic equations involving measures

2000 ◽  
Vol 130 (1) ◽  
pp. 167-187 ◽  
Author(s):  
L. Orsina ◽  
A. Prignet
Keyword(s):  
Elliptic Equations ◽  
Dirichlet Boundary ◽  
Existence Of Solutions ◽  
Model Problem ◽  
Nonlinear Elliptic Equations ◽  
Dirichlet Boundary Conditions ◽  
Regular Functions ◽  
Nonlinear Elliptic ◽  
Bounded Open Subset ◽  
Image Position

In this paper, we study the non-existence of solutions for the following (model) problem in a bounded open subset Ω of RN: with Dirichlet boundary conditions, where p > 1, q > 1 and μ is a bounded Radon measure. We prove that if λ is a measure which is concentrated on a set of zero r capacity (p < r ≤ N), and if q > r (p − 1)/(r − p), then there is no solution to the above problem, in the sense that if one approximates the measure λ with a sequence of regular functions fn, and if un is the sequence of solutions of the corresponding problems, then un converges to zero.We also study the non-existence of solutions for the bilateral obstacle problem with datum a measure λ concentrated on a set of zero p capacity, with u in for every υ in K, finding again that the only solution obtained by approximation is u = 0.

Multiple Positive Solutions For a Class of Nonlinear Elliptic Equations

2006 ◽  
Vol 6 (1) ◽  
Author(s):  
Shenghua Weng ◽  
Yongqing Li
Keyword(s):  
Boundary Value Problems ◽  
Positive Solutions ◽  
Elliptic Equations ◽  
Dirichlet Boundary ◽  
Nonlinear Elliptic Equations ◽  
Multiple Positive Solutions ◽  
Combined Effects ◽  
Multiplicity Results ◽  
Nonlinear Elliptic ◽  
Existence And Multiplicity

AbstractThis paper deals with a class of nonlinear elliptic Dirichlet boundary value problems where the combined effects of a sublinear and a superlinear term allow us to establish some existence and multiplicity results.

Padé approximant bounds on the positive solutions of some nonlinear elliptic equations

1978 ◽  
Vol 80 (3-4) ◽  
pp. 183-200
Author(s):  
M. F. Barnsley ◽  
D. Bessis
Keyword(s):  
Laplace Transform ◽  
Elliptic Equations ◽  
Positive Measure ◽  
Dirichlet Boundary ◽  
Homogeneous Equation ◽  
Nonlinear Elliptic Equations ◽  
Dirichlet Boundary Conditions ◽  
Upper And Lower Bounds ◽  
Nonlinear Elliptic ◽  
Positive Functions

SynopsisWe consider the equation Lφ − λpφ + γqφ2 = f on a bounded domain in Rn with homogeneous Neumann-Dirichlet boundary conditions. L is a negative definite uniformly elliptic differential operator, while, p, q and f are positive functions. We show that there exists exactly one positive solution for each λ ∈ R and γ > 0. This solution can be analytically continued throughout Re γ > 0: it is a Laplace transform of a positive measure. The measure is bounded prior to the bifurcation point of the associated “homogeneous” equation and unbounded after. Noting that any Laplace transform of positive measure has associated with it a natural sequence of Tchebycheff systems, it now follows that one can obtain monotonically converging upper and lower bounds which are provided by the generalized Padé approximants generated from the Tchebycheff systems.

scholarly journals Uniqueness of weak solution for nonlinear elliptic equations in divergence form

2000 ◽  
Vol 23 (5) ◽  
pp. 313-318 ◽  
Author(s):  
Xu Zhang
Keyword(s):  
Weak Solution ◽  
Weak Solutions ◽  
Elliptic Equations ◽  
Nonlinear Elliptic Equations ◽  
Divergence Form ◽  
Uniqueness Result ◽  
Quasilinear Elliptic Equations ◽  
Nonlinear Elliptic ◽  
Quasilinear Elliptic

We study the uniqueness of weak solutions for quasilinear elliptic equations in divergence form. Some counterexamples are given to show that our uniqueness result cannot be improved in the general case.

scholarly journals On a Class of Anisotropic Nonlinear Elliptic Equations with Variable Exponent

2013 ◽  
Vol 2013 ◽  
pp. 1-9
Author(s):  
Guoqing Zhang ◽  
Hongtao Zhang
Keyword(s):  
Weak Solution ◽  
Elliptic Equations ◽  
Existence And Uniqueness ◽  
Variable Exponent ◽  
Nonlinear Elliptic Equations ◽  
Nonlinear Elliptic ◽  
Priori Estimates

Based on truncation technique and priori estimates, we prove the existence and uniqueness of weak solution for a class of anisotropic nonlinear elliptic equations with variable exponentp(x)→growth. Furthermore, we also obtain that the weak solution is locally bounded and regular; that is, the weak solution isLloc∞(Ω)andC1,α(Ω).

On the Asymptotics of Solutions of Elliptic Equations in a Neighborhood of a Crack with Nonsmooth Front

2003 ◽  
Vol 10 (3) ◽  
pp. 543-548
Author(s):  
V. A. Kondrat'ev ◽  
V. A. Nikishkin
Keyword(s):  
Boundary Value Problem ◽  
Elliptic Equations ◽  
Dirichlet Boundary ◽  
Boundary Value ◽  
Second Order ◽  
Dirichlet Boundary Value Problem ◽  
Asymptotics Of Solutions ◽  
Divergent Form ◽  
Second Order Elliptic Equations

Abstract Two terms of asymptotics near crack are obtained for solutions of the Dirichlet boundary value problem for second-order elliptic equations in divergent form. The front of a crack is from 𝐶1+𝑠 and the coefficients of the equations belong to 𝐶𝑠 (0.5 < 𝑠 < 1).

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