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.

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.

Symmetry properties of positive solutions of elliptic equations in an infinite sectorial cone

1992 ◽  
Vol 122 (1-2) ◽  
pp. 137-160
Author(s):  
Chie-Ping Chu ◽  
Hwai-Chiuan Wang
Keyword(s):  
Boundary Conditions ◽  
Positive Solution ◽  
Positive Solutions ◽  
Elliptic Equations ◽  
Neumann Boundary ◽  
Neumann Boundary Conditions ◽  
Semilinear Elliptic Equations ◽  
Spherically Symmetric ◽  
Semilinear Elliptic ◽  
Symmetry Properties

SynopsisWe prove symmetry properties of positive solutions of semilinear elliptic equations Δu + f(u) = 0 with Neumann boundary conditions in an infinite sectorial cone. We establish that any positive solution u of such equations in an infinite sectorial cone ∑α in ℝ3 is spherically symmetric if the amplitude α of ∑α is not greater than π.

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

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.

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.

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.

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