A Note on a Comparison Result for Elliptic Equations

1977 ◽  
Vol 29 (5) ◽  
pp. 1081-1085 ◽  
Author(s):  
W. Allegretto
Keyword(s):  
Bounded Domain ◽  
Positive Solutions ◽  
Comparison Theorem ◽  
Elliptic Equations ◽  
Comparison Result ◽  
Mixed Derivatives ◽  
Image Position ◽  
Derivatives Of

In a recent paper [2], Bushard established and applied a comparison theorem for positive solutions to the equation:in an arbitrary bounded domain D of Euclidean w-space Rn. The proof of these results depended on the absence of mixed derivatives of u in the equation considered.

Positive solutions for non-homogeneous semilinear elliptic equations with data that changes sign

2003 ◽  
Vol 133 (2) ◽  
pp. 297-306 ◽  
Author(s):  
Qiuyi Dai ◽  
Yonggeng Gu
Keyword(s):  
Bounded Domain ◽  
Positive Solutions ◽  
Nonlinear Problem ◽  
Elliptic Equations ◽  
Linear Problem ◽  
Semilinear Elliptic Equations ◽  
Semilinear Elliptic ◽  
Existence Of Positive Solutions ◽  
Image Position

Let Ω ⊂ RN be a bounded domain. We consider the nonlinear problem and prove that the existence of positive solutions of the above nonlinear problem is closely related to the existence of non-negative solutions of the following linear problem: .In particular, if p > (N + 2)/(N − 2), then the existence of positive solutions of nonlinear problem is equivalent to the existence of non-negative solutions of the linear problem (for more details, we refer to theorems 1.2 and 1.3 in § 1 of this paper).

Multiple positive solutions of nonhomogeneous semilinear elliptic equations in ℝN

1996 ◽  
Vol 126 (2) ◽  
pp. 443-463 ◽  
Author(s):  
Cao Dao-Min ◽  
Zhou Huan-Song
Keyword(s):  
Positive Solutions ◽  
Elliptic Equations ◽  
Semilinear Elliptic Equations ◽  
Multiple Positive Solutions ◽  
Semilinear Elliptic ◽  
Sobolev Constant ◽  
Image Position

We consider the following problemwhere for all ≦f(x,u)≦c1up-1 + c2u for all x ∈ℝN,u≧0 with c1>0,c2∈(0, 1), 2<p<(2N/(N – 2)) if N ≧ 3, 2 ≧ + ∝ if N = 2. We prove that (*) has at least two positive solutions ifand h≩0 in ℝN, where S is the best Sobolev constant and

On some nonlinear elliptic equations involving derivatives of the nonlinearity

1985 ◽  
Vol 100 (3-4) ◽  
pp. 281-294 ◽  
Author(s):  
J. Carrillo ◽  
M. Chipot
Keyword(s):  
Boundary Value Problems ◽  
Elliptic Equations ◽  
Existence And Uniqueness ◽  
Elliptic Boundary ◽  
Nonlinear Elliptic Equations ◽  
Elliptic Boundary Value Problems ◽  
Elliptic Boundary Value ◽  
Nonlinear Elliptic ◽  
Image Position ◽  
Derivatives Of

SynopsisWe give some results on existence and uniqueness for the solution of elliptic boundary value problems of typewhen the βi are not necessarily smooth.

Liouville-type results for positive solutions of pseudo-relativistic Schrödinger system

2021 ◽  
pp. 1-33
Author(s):  
Yuxia Guo ◽  
Shaolong Peng
Keyword(s):  
Bounded Domain ◽  
Positive Solutions ◽  
Unbounded Domain ◽  
Direct Method ◽  
Strict Monotonicity ◽  
Liouville Type ◽  
Moving Plane ◽  
Liouville Type Results ◽  
Image Position ◽  
Schrödinger System

In this paper, we are concerned with the physically engaging pseudo-relativistic Schrödinger system: \[ \begin{cases} \left(-\Delta+m^{2}\right)^{s}u(x)=f(x,u,v,\nabla u) & \hbox{in } \Omega,\\ \left(-\Delta+m^{2}\right)^{t}v(x)=g(x,u,v,\nabla v) & \hbox{in } \Omega,\\ u>0,v>0 & \hbox{in } \Omega, \\ u=v\equiv 0 & \hbox{in } \mathbb{R}^{N}\setminus\Omega, \end{cases} \] where $s,t\in (0,1)$ and the mass $m>0.$ By using the direct method of moving plane, we prove the strict monotonicity, symmetry and uniqueness for positive solutions to the above system in a bounded domain, unbounded domain, $\mathbb {R}^{N}$ , $\mathbb {R}^{N}_{+}$ and a coercive epigraph domain $\Omega$ in $\mathbb {R}^{N}$ , respectively.

Solutions with multiple peaks for nonlinear elliptic equations

1999 ◽  
Vol 129 (2) ◽  
pp. 235-264 ◽  
Author(s):  
Daomin Cao ◽  
Ezzat S. Noussair ◽  
Shusen Yan
Keyword(s):  
Positive Solutions ◽  
Critical Points ◽  
Elliptic Equations ◽  
Nonlinear Elliptic Equations ◽  
Multiple Peaks ◽  
Nonlinear Elliptic ◽  
Image Position

Solutions with peaks near the critical points of Q(x) are constructed for the problemWe establish the existence of 2k −1 positive solutions when Q(x) has k non-degenerate critical points in ℝN

Universal inequalities for eigenvalues of a system of elliptic equations

2009 ◽  
Vol 139 (2) ◽  
pp. 273-285 ◽  
Author(s):  
Qing-Ming Cheng ◽  
Hong-Cang Yang
Keyword(s):  
Euclidean Space ◽  
Eigenvalue Problem ◽  
Bounded Domain ◽  
Lower Order ◽  
Elliptic Equations ◽  
Elastic Vibration ◽  
Dimensional Euclidean Space ◽  
Explicit Method ◽  
Universal Inequalities ◽  
Image Position

Let D be a bounded domain in an n-dimensional Euclidean space ℝn. Assume thatare eigenvalues of an eigenvalue problem of a system of n elliptic equations:In particular, when n=3, the eigenvalue problem describes the behaviour of the elastic vibration. We obtain universal inequalities for eigenvalues of the above eigenvalue problem by making use of a direct and explicit method; our results are sharper than one of Hook. Furthermore, a universal inequality for lower-order eigenvalues of the above eigenvalue problem is also derived.

Non-homogeneous elliptic equations in exterior domains

2006 ◽  
Vol 136 (1) ◽  
pp. 139-147 ◽  
Author(s):  
J. do Ó ◽  
S. Lorca ◽  
J. Sánchez ◽  
P. Ubilla
Keyword(s):  
Elliptic Equation ◽  
Positive Solutions ◽  
Elliptic Equations ◽  
Exterior Domains ◽  
Existence And Multiplicity ◽  
Negative Function ◽  
Multiplicity Of Positive Solutions ◽  
Image Position ◽  
Typical Model ◽  
Homogeneous Elliptic Equation

We study the existence and multiplicity of positive solutions of the non-homogeneous elliptic equation where N ≥ 3, the nonlinearity f is superlinear at both zero and infinity, q is a non-trivial, non-negative function, and a and b are non-negative parameters. A typical model is given by f(u) = up, with p ≥ 1.

Decaying positive solutions to a system of nonlinear elliptic equations

2006 ◽  
Vol 136 (2) ◽  
pp. 301-320
Author(s):  
Fenfei Chen ◽  
Miaoxin Yao
Keyword(s):  
Positive Solution ◽  
Elliptic System ◽  
Positive Solutions ◽  
Elliptic Equations ◽  
Nonlinear Elliptic Equations ◽  
Second Order ◽  
Nonlinear Elliptic System ◽  
Nonlinear Elliptic ◽  
Image Position ◽  
Bounded Below

In this paper, the second-order nonlinear elliptic system with α, γ < 1 and β ≥ 1, is considered in RN, N ≥ 3. Under suitable hypotheses on functions fi, gi, hi (i = 1, 2) and P, it is shown that this system possesses an entire positive solution , 0 < θ < 1, such that both u and v are bounded below and above by constant multiples of |x|2−N for all |x| ≥ 1.

A perturbation result of semilinear elliptic equations in exterior strip domains

1997 ◽  
Vol 127 (5) ◽  
pp. 983-1004 ◽  
Author(s):  
Tsing-san Hsu ◽  
Hwai-chiuan Wang
Keyword(s):  
Dirichlet Problem ◽  
Positive Solutions ◽  
Elliptic Equations ◽  
Semilinear Elliptic Equations ◽  
Energy Solution ◽  
Perturbation Result ◽  
Semilinear Elliptic ◽  
Image Position

SynopsisIn this paper we show that if the decay of nonzero ƒ is fast enough, then the perturbation Dirichlet problem −Δu + u = up + ƒ(z) in Ω has at least two positive solutions, wherea bounded C1,1 domain S = × ω Rn, D is a bounded C1,1 domain in Rm+n such that D ⊂⊂ S and Ω = S\D. In case ƒ ≡ 0, we assert that there is a positive higher-energy solution providing that D is small.

Asymptotic behaviour and symmetry of positive solutions to nonlinear elliptic equations in a half-space

2016 ◽  
Vol 146 (6) ◽  
pp. 1243-1263 ◽  
Author(s):  
Lei Wei
Keyword(s):  
Half Space ◽  
Positive Solutions ◽  
Elliptic Equations ◽  
Nonlinear Elliptic Equations ◽  
Unique Positive Solution ◽  
Bounded Smooth Domain ◽  
Nonlinear Elliptic ◽  
Bounded Smooth ◽  
Image Position ◽  
General Nonlinearity

We consider the following equation:where d(x) = d(x, ∂Ω), θ > –2 and Ω is a half-space. The existence and non-existence of several kinds of positive solutions to this equation when , f(u) = up(p > 1) and Ω is a bounded smooth domain were studied by Bandle, Moroz and Reichel in 2008. Here, we study exact the behaviour of positive solutions to this equation as d(x) → 0+ and d(x) → ∞, respectively, and the symmetry of positive solutions when , Ω is a half-space and f(u) is a more general nonlinearity term than up. Under suitable conditions for f, we show that the equation has a unique positive solution W, which is a function of x1 only, and W satisfies

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