Radial solutions of a semilinear elliptic equation at a critical exponent

1988 ◽  
Vol 104 (3) ◽  
pp. 251-270 ◽  
Author(s):  
Christopher K. R. T. Jones
Keyword(s):  
Elliptic Equation ◽  
Critical Exponent ◽  
Semilinear Elliptic Equation ◽  
Radial Solutions ◽  
Semilinear Elliptic

Multi-bump solutions of -Δn = K(x)u (n+2)/(n-2) on lattices in ℝ n

2018 ◽  
Vol 2018 (743) ◽  
pp. 163-211 ◽  
Author(s):  
Yanyan Li ◽  
Juncheng Wei ◽  
Haoyuan Xu
Keyword(s):  
Elliptic Equation ◽  
Critical Exponent ◽  
Critical Point ◽  
Semilinear Elliptic Equation ◽  
Natural Conditions ◽  
Semilinear Elliptic ◽  
Infinite Lattices

Abstract We consider the following semilinear elliptic equation with critical exponent: Δ u = K(x) u^{(n+2)/(n-2)} , u > 0 in \mathbb{R}^{n} , where {n\geq 3} , {K>0} is periodic in ( x_{1} ,…, x_{k} ) with 1 \leq k < (n-2)/2. Under some natural conditions on K near a critical point, we prove the existence of multi-bump solutions where the centers of bumps can be placed in some lattices in {\mathbb{R}^{k}} , including infinite lattices. We also show that for k \geq (n-2)/2, no such solutions exist.

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.

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