Local estimates for a semilinear elliptic equation with Sobolev critical exponent and application to a uniqueness result

2001 ◽  
Vol 8 (3) ◽  
pp. 251-283 ◽  
Author(s):  
Katiuscia Cerqueti ◽  
Massimo Grossi
Keyword(s):  
Elliptic Equation ◽  
Critical Exponent ◽  
Semilinear Elliptic Equation ◽  
Uniqueness Result ◽  
Semilinear Elliptic ◽  
Sobolev Critical Exponent

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.

scholarly journals A uniqueness result for some singular semilinear elliptic equations

2016 ◽  
Vol 18 (06) ◽  
pp. 1550084 ◽  
Author(s):  
Annamaria Canino ◽  
Berardino Sciunzi
Keyword(s):  
Elliptic Equation ◽  
Boundary Conditions ◽  
Open Subset ◽  
Elliptic Equations ◽  
Dirichlet Boundary ◽  
Semilinear Elliptic Equation ◽  
Uniqueness Result ◽  
Dirichlet Boundary Conditions ◽  
Semilinear Elliptic ◽  
Bounded Open Subset

Given [Formula: see text] a bounded open subset of [Formula: see text], we consider non-negative solutions to the singular semilinear elliptic equation [Formula: see text] in [Formula: see text], under zero Dirichlet boundary conditions. For [Formula: see text] and [Formula: see text], we prove that the solution is unique.

scholarly journals Positive radial symmetric solutions for a class of elliptic problems with critical exponent and -1 growth

2021 ◽  
Vol 10 (1) ◽  
pp. 1222-1234
Author(s):  
Chun-Yu Lei ◽  
Jia-Feng Liao
Keyword(s):  
Elliptic Equation ◽  
Critical Exponent ◽  
Critical Point ◽  
Critical Point Theory ◽  
Semilinear Elliptic Equation ◽  
Point Theory ◽  
Moving Plane Method ◽  
Plane Method ◽  
Semilinear Elliptic ◽  
Moving Plane

Abstract In this paper, we consider a class of semilinear elliptic equation with critical exponent and -1 growth. By using the critical point theory for nonsmooth functionals, two positive solutions are obtained. Moreover, the symmetry and monotonicity properties of the solutions are proved by the moving plane method. Our results improve the corresponding results in the literature.

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