scholarly journals Asymptotic behavior of the least-energy solutions of a semilinear elliptic equation with the Hardy–Sobolev critical exponent

2017 ◽  
Vol 262 (3) ◽  
pp. 3107-3131 ◽  
Author(s):  
Masato Hashizume
Keyword(s):  
Asymptotic Behavior ◽  
Elliptic Equation ◽  
Critical Exponent ◽  
Semilinear Elliptic Equation ◽  
Semilinear Elliptic ◽  
Least Energy Solutions ◽  
Sobolev Critical Exponent ◽  
Least Energy

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.

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