Entire Large Solutions to Elliptic Equations of Power Non-linearities with Variable Exponents

2013 ◽  
Vol 13 (3) ◽  
Author(s):  
Alan V. Lair ◽  
Ahmed Mohammed
Keyword(s):  
Elliptic Equation ◽  
Positive Solution ◽  
Elliptic Equations ◽  
Variable Exponent ◽  
Variable Exponents ◽  
Large Solutions ◽  
Existence And Nonexistence

AbstractWe give conditions on the variable exponent q that ensure the existence and nonexistence of a positive solution to the elliptic equation Δu = u

Nonlinear Elliptic Equations in Strip-Like Domains

2012 ◽  
Vol 12 (4) ◽  
Author(s):  
Jaeyoung Byeon ◽  
Kazunaga Tanaka
Keyword(s):  
Elliptic Equation ◽  
Positive Solution ◽  
Bounded Domain ◽  
Elliptic Equations ◽  
Nonlinear Elliptic Equation ◽  
Nonlinear Elliptic Equations ◽  
Nonlinear Elliptic

AbstractWe study the existence of a positive solution of a nonlinear elliptic equationwhere k ≥ 2 and D is a bounded domain domain in R

scholarly journals EXISTENCE OF POSITIVE EVANESCENT SOLUTIONS TO SOME QUASILINEAR ELLIPTIC EQUATIONS

2008 ◽  
Vol 78 (1) ◽  
pp. 157-162 ◽  
Author(s):  
OCTAVIAN G. MUSTAFA
Keyword(s):  
Elliptic Equation ◽  
Positive Solution ◽  
Elliptic Equations ◽  
Exterior Domain ◽  
Quasilinear Elliptic Equations ◽  
Image Position ◽  
Quasilinear Elliptic

AbstractWe establish that the elliptic equation defined in an exterior domain of ℝn, n≥3, has a positive solution which decays to 0 as $\vert x\vert \rightarrow +\infty $ under quite general assumptions upon f and g.

On the removability of isolated singular points for elliptic equations involving variable exponent

2016 ◽  
Vol 5 (2) ◽  
Author(s):  
Yongqiang Fu ◽  
Yingying Shan
Keyword(s):  
Singular Point ◽  
Elliptic Equations ◽  
Variable Exponent ◽  
Singular Points ◽  
Sufficient Condition ◽  
Variable Exponents ◽  
Isolated Singularities

AbstractIn this paper, we study the problem of removable isolated singularities for elliptic equations with variable exponents. We give a sufficient condition for removability of the isolated singular point for the equations in

A Singular Semilinear Elliptic Equation with a Variable Exponent

2016 ◽  
Vol 16 (3) ◽  
Author(s):  
José Carmona ◽  
Pedro J. Martínez-Aparicio
Keyword(s):  
Elliptic Equation ◽  
Elliptic Equations ◽  
Variable Exponent ◽  
Model Problem ◽  
Semilinear Elliptic Equation ◽  
Semilinear Elliptic Equations ◽  
Bounded Set ◽  
Semilinear Elliptic

AbstractIn this paper we consider singular semilinear elliptic equations with a variable exponent whose model problem isHere Ω is an open bounded set of

scholarly journals The Solvability of Fractional Elliptic Equation with the Hardy Potential

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-8
Author(s):  
Siyu Gao ◽  
Shuibo Huang ◽  
Qiaoyu Tian ◽  
Zhan-Ping Ma
Keyword(s):  
Elliptic Equation ◽  
Lipschitz Domain ◽  
Sobolev Inequality ◽  
Elliptic Equations ◽  
Sharp Constant ◽  
Hardy Potential ◽  
Bounded Lipschitz Domain ◽  
Existence And Nonexistence ◽  
Nonexistence Of Solutions ◽  
Fractional Laplace Operator

In this paper, we study the existence and nonexistence of solutions to fractional elliptic equations with the Hardy potential −Δsu−λu/x2s=ur−1+δgu,in Ω,ux>0,in Ω,ux=0,in ℝN∖Ω, where Ω⊂ℝN is a bounded Lipschitz domain with 0∈Ω, −Δs is a fractional Laplace operator, s∈0,1, N>2s, δ is a positive number, 2<r<rλ,s≡N+2s−2αλ/N−2s−2αλ+1, αλ∈0,N−2s/2 is a parameter depending on λ, 0<λ<ΛN,s, and ΛN,s=22sΓ2N+2s/4/Γ2N−2s/4 is the sharp constant of the Hardy–Sobolev inequality.

scholarly journals On Existence of Solution for a Class of Semilinear Elliptic Equations with Nonlinearities That Lies between Different Powers

2008 ◽  
Vol 2008 ◽  
pp. 1-6 ◽  
Author(s):  
Claudianor O. Alves ◽  
Marco A. S. Souto
Keyword(s):  
Elliptic Equation ◽  
Positive Solution ◽  
Elliptic Equations ◽  
Semilinear Elliptic Equation ◽  
Existence Of Solution ◽  
Semilinear Elliptic Equations ◽  
Semilinear Elliptic

We prove that the semilinear elliptic equation−Δu=f(u), inΩ,u=0, on∂Ωhas a positive solution when the nonlinearityfbelongs to a class which satisfiesμtq≤f(t)≤Ctpat infinity and behaves liketqnear the origin, where1<q<(N+2)/(N−2)ifN≥3and1<q<+∞ifN=1,2. In our approach, we do not need the Ambrosetti-Rabinowitz condition, and the nonlinearity does not satisfy any hypotheses such those required by the blowup method. Furthermore, we do not impose any restriction on the growth ofp.

scholarly journals Variable Anisotropic Hardy Spaces with Variable Exponents

2021 ◽  
Vol 9 (1) ◽  
pp. 65-89
Author(s):  
Zhenzhen Yang ◽  
Yajuan Yang ◽  
Jiawei Sun ◽  
Baode Li
Keyword(s):  
Hardy Spaces ◽  
Variable Exponent ◽  
Atomic Decomposition ◽  
Integral Operators ◽  
Moment Condition ◽  
Variable Exponents ◽  
Hölder Continuous ◽  
Multi Level ◽  
The Moment ◽  
Variable Anisotropy

Abstract Let p(·) : ℝ n → (0, ∞] be a variable exponent function satisfying the globally log-Hölder continuous and let Θ be a continuous multi-level ellipsoid cover of ℝ n introduced by Dekel et al. [12]. In this article, we introduce highly geometric Hardy spaces Hp (·)(Θ) via the radial grand maximal function and then obtain its atomic decomposition, which generalizes that of Hardy spaces Hp (Θ) on ℝ n with pointwise variable anisotropy of Dekel et al. [16] and variable anisotropic Hardy spaces of Liu et al. [24]. As an application, we establish the boundedness of variable anisotropic singular integral operators from Hp (·)(Θ) to Lp (·)(ℝ n ) in general and from Hp (·)(Θ) to itself under the moment condition, which generalizes the previous work of Bownik et al. [6] on Hp (Θ).

scholarly journals Large solutions of semilinear elliptic equations with nonlinear gradient terms

1999 ◽  
Vol 22 (4) ◽  
pp. 869-883 ◽  
Author(s):  
Alan V. Lair ◽  
Aihua W. Wood
Keyword(s):  
Positive Solutions ◽  
Compact Support ◽  
Elliptic Equations ◽  
Nonnegative Function ◽  
Semilinear Elliptic Equations ◽  
Decay Condition ◽  
Large Solutions ◽  
Semilinear Elliptic ◽  
The Given

We show that large positive solutions exist for the equation(P±):Δu±|∇u|q=p(x)uγinΩ⫅RN(N≥3)for appropriate choices ofγ>1,q>0in which the domainΩis either bounded or equal toRN. The nonnegative functionpis continuous and may vanish on large parts ofΩ. IfΩ=RN, thenpmust satisfy a decay condition as|x|→∞. For(P+), the decay condition is simply∫0∞tϕ(t)dt<∞, whereϕ(t)=max|x|=tp(x). For(P−), we require thatt2+βϕ(t)be bounded above for some positiveβ. Furthermore, we show that the given conditions onγandpare nearly optimal for equation(P+)in that no large solutions exist if eitherγ≤1or the functionphas compact support inΩ.

Positive Solutions for Elliptic Equations With Supercritical Nonlinearity

2009 ◽  
Vol 9 (3) ◽  
Author(s):  
Paulo Rabelo
Keyword(s):  
Elliptic Equation ◽  
Elliptic Equations ◽  
A Priori Estimates ◽  
A Priori ◽  
Auxiliary Problem ◽  
Semilinear Elliptic Equation ◽  
Minimax Methods ◽  
Semilinear Elliptic ◽  
Priori Estimates ◽  
Supercritical Nonlinearity

AbstractIn this paper minimax methods are employed to establish the existence of a bounded positive solution for semilinear elliptic equation of the form−∆u + V (x)u = P(x)|u|where the nonlinearity has supercritical growth and the potential can change sign. The solutions of the problem above are obtained by proving a priori estimates for solutions of a suitable auxiliary problem.

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