Some results for semilinear elliptic equations with critical potential

2002 ◽  
Vol 132 (1) ◽  
pp. 1-24 ◽  
Author(s):  
B. Abdellaoui ◽  
I. Peral
Keyword(s):  
Elliptic Equations ◽  
Elliptic Problems ◽  
Semilinear Elliptic Equations ◽  
Critical Potential ◽  
Semilinear Elliptic ◽  
Image Position

This paper is devoted to the study of the elliptic problems with a critical potential, where N ≥ 3, λ ≥ 0 and 0 < q < 1 < p ≤ (N + 2)/(N − 2). Existence, multiplicity, behaviour in x = 0 and bifurcation are considered under some hypotheses in h and g.

Existence and non-existence results for semilinear elliptic problems in unbounded domains

1982 ◽  
Vol 93 (1-2) ◽  
pp. 1-14 ◽  
Author(s):  
M. J. Esteban ◽  
P. L. Lions
Keyword(s):  
Elliptic Equations ◽  
Elliptic Problems ◽  
Unbounded Domains ◽  
Existence Results ◽  
Semilinear Elliptic Equations ◽  
Semilinear Elliptic ◽  
Semilinear Elliptic Problems ◽  
Image Position

SynopsisIn this paper, we prove various existence and non-existence results for semilinear elliptic problems in unbounded domains. In particular we prove for general classes of unbounded domains that there exists no solution distinct from 0 offor any smooth f satisfying f(0) = 0. This result is obtained by the use of new identities that solutions of semilinear elliptic equations satisfy.

scholarly journals Positive Solutions for Elliptic Problems with the Nonlinearity Containing Singularity and Hardy-Sobolev Exponents

2020 ◽  
Vol 2020 ◽  
pp. 1-7
Author(s):  
Yong-Yi Lan ◽  
Xian Hu ◽  
Bi-Yun Tang
Keyword(s):  
Variational Methods ◽  
Positive Solutions ◽  
Elliptic Equations ◽  
Elliptic Problems ◽  
Semilinear Elliptic Equations ◽  
Existence And Multiplicity ◽  
Semilinear Elliptic ◽  
Multiplicity Of Positive Solutions

In this paper, we study multiplicity of positive solutions for a class of semilinear elliptic equations with the nonlinearity containing singularity and Hardy-Sobolev exponents. Using variational methods, we establish the existence and multiplicity of positive solutions for the problem.

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

Multiple solutions for semilinear elliptic equations in unbounded cylinder domains

2004 ◽  
Vol 134 (4) ◽  
pp. 719-731 ◽  
Author(s):  
Tsing-San Hsu
Keyword(s):  
Dirichlet Problem ◽  
Positive Solution ◽  
Elliptic Equations ◽  
Multiple Solutions ◽  
Semilinear Elliptic Equations ◽  
Semilinear Elliptic ◽  
Unbounded Cylinder ◽  
Image Position

In this paper, we show that if b(x) ≥ b∞ > 0 in Ω̄ and there exist positive constants C, δ, R0 such that where x = (y, z) ∈ RN with y ∈ Rm, z ∈ Rn, N = m + n ≥ 3, m ≥ 2, n ≥ 1, 1 < p < (N + 2)/(N − 2), ω ⊆ Rm a bounded C1,1 domain and Ω = ω × Rn, then the Dirichlet problem −Δu + u = b(x)|u|p−1u in Ω has a solution that changes sign in Ω, in addition to a positive solution.

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.

The existence of a positive solution of semilinear elliptic equations with limiting Sobolev exponent

1991 ◽  
Vol 117 (1-2) ◽  
pp. 75-88 ◽  
Author(s):  
Shixiao Wang
Keyword(s):  
Positive Solution ◽  
Elliptic Equations ◽  
Semilinear Elliptic Equations ◽  
Thin Domain ◽  
Existence Of A Solution ◽  
Semilinear Elliptic ◽  
Sobolev Exponent ◽  
Image Position

SynopsisOur paper concerns the existence of a positive solution for the equation:A new condition, which guarantees the existence of a solution of the above equation, has been established. It has also given some sharp information in the cases where: (1) a(x) = λ = const. and Ω is a “thin” domain; (2) Ω is a ball and a(x) is a radially symmetrical function.

Radial entire solutions to even order semilinear elliptic equations in the plane

1987 ◽  
Vol 105 (1) ◽  
pp. 275-287 ◽  
Author(s):  
Takaŝi Kusano ◽  
Manabu Naito ◽  
Charles A. Swanson
Keyword(s):  
Elliptic Equations ◽  
Sufficient Conditions ◽  
Semilinear Elliptic Equations ◽  
Two Dimensional ◽  
Entire Solutions ◽  
Semilinear Elliptic ◽  
Negative Constant ◽  
Even Order ◽  
Image Position ◽  
Necessary And Sufficient

SynopsisSemilinear elliptic equations of the formare considered, where Δm is the m-th iterate of the two-dimensional Laplacian Δ, p(t) is continuous in [0, ∞), and f(u is continuous and positive either in (0, ∞) or in ℝ.Our main objective is to present conditions on p and f which imply the existence of radial entire solutions to (*), that is, those functions of class C2m(ℝ2) which depend only on |x| and satisfy (*) pointwise in ℝ2.First, necessary and sufficient conditions are established for equation (*), with p(t) > 0 in [0, ∞), to possess infinitely many positive radial entire solutions which are asymptotic to positive constant multiples of |x|2m−2 log |x as |x| → ∞. Secondly, it is shown that, in the case p(t < 0, in [ 0, ∞) and f(u) > 0 is nondecreasing in ℝ, equation (*) always has eventually negative radial entire solutions, all of which decrease at least as fast as negative constant multiples of |x|2m−2 log |x| as |x| → ∞. Our results seem to be new even when specialised to the prototypeswhere γ is a constant.

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).

scholarly journals Boundedness of Stable Solutions to Semilinear Elliptic Equations: A Survey

2017 ◽  
Vol 17 (2) ◽  
Author(s):  
Xavier Cabré
Keyword(s):  
Elliptic Equations ◽  
Elliptic Problems ◽  
Semilinear Elliptic Equations ◽  
Semilinear Elliptic ◽  
Semilinear Elliptic Problems ◽  
Stable Solutions

AbstractThis article is a survey on boundedness results for stable solutions to semilinear elliptic problems. For these solutions, we present the currently known

Sign in / Sign up

Export Citation Format

Share Document