scholarly journals Boundary singularities on a wedge-like domain of a semilinear elliptic equation

2015 ◽  
Vol 145 (5) ◽  
pp. 979-1006 ◽  
Author(s):  
Konstantinos T. Gkikas
Keyword(s):  
Elliptic Equation ◽  
Weak Solutions ◽  
Semilinear Elliptic Equation ◽  
Fast Decay ◽  
Semilinear Elliptic ◽  
Image Position

Letn≥ 2 and letbe a Lipschitz wedge-like domain. We construct positive weak solutions of the problemthat vanish in a suitable trace sense on∂Ω, but which are singular at a prescribed ‘edge’ ofΩifpis equal to or slightly above a certain exponentp0> 1 that depends onΩ. Moreover, for the case in whichΩis unbounded, the solutions have fast decay at infinity.

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.

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.

WEAK SOLUTIONS OF QUASILINEAR BIHARMONIC PROBLEMS WITH POSITIVE, INCREASING AND CONVEX NONLINEARITIES

2008 ◽  
Vol 06 (03) ◽  
pp. 213-227 ◽  
Author(s):  
I. ABID ◽  
M. JLELI ◽  
N. TRABELSI
Keyword(s):  
Elliptic Equation ◽  
Boundary Conditions ◽  
Fourth Order ◽  
Weak Solutions ◽  
Source Term ◽  
Semilinear Elliptic Equation ◽  
Laplacian Operator ◽  
Navier Boundary Conditions ◽  
Semilinear Elliptic ◽  
Biharmonic Problems

We study the existence of positive weak solutions to a fourth-order semilinear elliptic equation with Navier boundary conditions and a positive, increasing and convex source term. We also prove the uniqueness of extremal solutions. In particular, we generalize results of Mironescu and Rădulescu for the bi-Laplacian operator.

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.

Multiple positive solutions for an unbounded Dirichlet boundary problem involving sign-changing weight

2009 ◽  
Vol 139 (6) ◽  
pp. 1297-1325
Author(s):  
Tsung-fang Wu
Keyword(s):  
Elliptic Equation ◽  
Positive Solutions ◽  
Dirichlet Boundary ◽  
Semilinear Elliptic Equation ◽  
Infinite Strip ◽  
Multiple Positive Solutions ◽  
Semilinear Elliptic ◽  
Multiplicity Of Positive Solutions ◽  
Image Position ◽  
Dirichlet Boundary Problem

We study the multiplicity of positive solutions for the following semilinear elliptic equation:where 1 < q < 2 < p < 2* (2* = 2N/(N − 2) if N ≥ 3, 2* = ∞ if N = 2), the parameters λ, μ ≥ 0, is an infinite strip in ℝN and Θ is a bounded domain in ℝN−1 We assume that fλ(x) = λf+(x) + f−(x) and gμ(x) = a(x) + μb(x), where the functions f±, a and b satisfy suitable conditions.

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