Semilinear elliptic equations of the Hénon-type in hyperbolic space

2016 ◽  
Vol 18 (02) ◽  
pp. 1550026 ◽  
Author(s):  
P. C. Carrião ◽  
L. F. O. Faria ◽  
O. H. Miyagaki
Keyword(s):  
Unit Ball ◽  
Positive Solution ◽  
Hyperbolic Space ◽  
Elliptic Equations ◽  
Mountain Pass Theorem ◽  
Semilinear Elliptic Equations ◽  
Mountain Pass ◽  
Sobolev Embedding ◽  
Radial Functions ◽  
Semilinear Elliptic

This paper deals with a class of the semilinear elliptic equations of the Hénon-type in hyperbolic space. The problem involves a logarithm weight in the Poincaré ball model, bringing singularities on the boundary. Considering radial functions, a compact Sobolev embedding result is proved, which extends a former Ni result made for a unit ball in [Formula: see text] Combining this compactness embedding with the Mountain Pass Theorem, a result of the existence of positive solution is established.

Decaying Solutions Of 2mth Order Elliptic Problems

1991 ◽  
Vol 43 (3) ◽  
pp. 449-460 ◽  
Author(s):  
W. Allegretto ◽  
L. S. Yu
Keyword(s):  
Positive Solution ◽  
Elliptic Problem ◽  
Mountain Pass Theorem ◽  
Elliptic Problems ◽  
Mountain Pass ◽  
Weighted Spaces ◽  
Semilinear Elliptic Problem ◽  
Semilinear Elliptic ◽  
Regularity Estimates ◽  
Open Question

AbstractWe consider a semilinear elliptic problem , (n > 2m). Under suitable conditions on f, we show the existence of a decaying positive solution. We do not employ radial arguments. Our main tools are weighted spaces, various applications of the Mountain Pass Theorem and LP regularity estimates of Agmon. We answer an open question of Kusano, Naito and Swanson [Canad. J. Math. 40(1988), 1281-1300] in the superlinear case: , and improve the results of Dalmasso [C. R. Acad. Sci. Paris 308(1989), 411-414] for the case .

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 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.

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.

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.

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