Morse index and symmetry breaking for an elliptic equation with negative exponent in expanding annuli

2017 ◽  
Vol 147 (6) ◽  
pp. 1215-1232
Author(s):  
Zongming Guo ◽  
Linfeng Mei ◽  
Zhitao Zhang
Keyword(s):  
Elliptic Equation ◽  
Symmetry Breaking ◽  
Morse Index ◽  
Blow Up ◽  
Semilinear Elliptic Equation ◽  
Radial Solutions ◽  
Entire Solutions ◽  
Semilinear Elliptic ◽  
Image Position

Bifurcation of non-radial solutions from radial solutions of a semilinear elliptic equation with negative exponent in expanding annuli of ℝ2 is studied. To obtain the main results, we use a blow-up argument via the Morse index of the regular entire solutions of the equationThe main results of this paper can be seen as applications of the results obtained recently for finite Morse index solutions of the equationwith N ⩾ 2 and p > 0.

A bifurcation diagram of solutions to an elliptic equation with exponential nonlinearity in higher dimensions

2017 ◽  
Vol 148 (1) ◽  
pp. 101-122 ◽  
Author(s):  
Hiroaki Kikuchi ◽  
Juncheng Wei
Keyword(s):  
Elliptic Equation ◽  
Bifurcation Diagram ◽  
Singular Solution ◽  
Morse Index ◽  
Higher Dimensions ◽  
Semilinear Elliptic Equation ◽  
Regular Solutions ◽  
Exponential Nonlinearity ◽  
Semilinear Elliptic ◽  
Image Position

We consider the following semilinear elliptic equation:where B1 is the unit ball in ℝd, d ≥ 3, λ > 0 and p > 0. Firstly, following Merle and Peletier, we show that there exists an eigenvalue λp,∞ such that (*) has a solution (λp,∞,Wp) satisfying lim|x|→0Wp(x) = ∞. Secondly, we study a bifurcation diagram of regular solutions to (*). It follows from the result of Dancer that (*) has an unbounded bifurcation branch of regular solutions that emanates from (λ, u) = (0, 0). Here, using the singular solution, we show that the bifurcation branch has infinitely many turning points around λp,∞ when 3 ≤ d ≤ 9. We also investigate the Morse index of the singular solution in the d ≥ 11 case.

scholarly journals On the blow-up and positive entire solutions of semilinear elliptic equations

2000 ◽  
Vol 43 (3) ◽  
pp. 625-631
Author(s):  
Jann-Long Chern
Keyword(s):  
Elliptic Equations ◽  
Blow Up ◽  
Type Equation ◽  
Semilinear Elliptic Equation ◽  
Complete Classification ◽  
Entire Solutions ◽  
Solution Structures ◽  
Semilinear Elliptic ◽  
Image Position

AbstractIn this paper we consider the following semilinear elliptic equationwhere n ≥ 3, and β ≥ 0, γ ≥ 0, q > p ≥ 1, μ and ν are real constants. We note that if γ = 0, β > 0 and ν ≥ 2, then the equation above is called the Matukuma-type equation. If β = 0, γ > 0 and ν > 2, then the complete classification of all possible positive solutions had been conducted by Cheng and Ni. If β > 0, γ > 0 and μ ≥ ν ≥ 2, then some results about the maximal solution and positive solution structures can be found in Chern. The purpose of this paper is to discuss and investigate the blow-up and positive entire solutions of the equation above for the μ ≥ 2 ≥ ν case.

On positive entire solutions of the elliptic equation Δu + K(x)up = 0 and its applications to Riemannian geometry

1996 ◽  
Vol 126 (2) ◽  
pp. 225-237 ◽  
Author(s):  
Changfeng Gui
Keyword(s):  
Elliptic Equation ◽  
Asymptotic Behaviour ◽  
Scalar Curvature ◽  
Riemannian Geometry ◽  
Semilinear Elliptic Equation ◽  
Entire Space ◽  
Entire Solutions ◽  
Curvature Equation ◽  
Semilinear Elliptic ◽  
Special Case

We study the existence and asymptotic behaviour of positive solutions of a semilinear elliptic equation in entire space. A special case of this equation is the scalar curvature equation which arises in Riemannian geometry.

Steady symmetric vortex pairs and rearrangements

1988 ◽  
Vol 108 (3-4) ◽  
pp. 269-290 ◽  
Author(s):  
G. R. Burton
Keyword(s):  
Elliptic Equation ◽  
A Priori ◽  
Vortex Pair ◽  
Semilinear Elliptic Equation ◽  
Symmetric Pair ◽  
Vorticity Field ◽  
Vortex Pairs ◽  
Semilinear Elliptic ◽  
Image Position ◽  
Increasing Function

SynopsisWe prove an existence theorem for a steady planar flow of an ideal fluid, containing a bounded symmetric pair of vortices, and approaching a uniform flow at infinity. The data prescribed are the rearrangement class of the vorticity field, and either the momentum impulse of the vortex pair, or the velocity of the vortex pair relative to the fluid at infinity. The stream function ψ for the flow satisfies the semilinear elliptic equationin a half-plane bounded by the line of symmetry, where φ is an increasing function that is unknown a priori. The results are proved by maximising the kinetic energy over all flows whose vorticity fields are rearrangements of a specified function.

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