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.

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.

scholarly journals Bifurcation of positive entire solutions for a semilinear elliptic equation

2005 ◽  
Vol 72 (3) ◽  
pp. 349-370
Author(s):  
Tsing-San Hsu ◽  
Huei-Li Lin
Keyword(s):  
Elliptic Equation ◽  
Continuous Function ◽  
Unique Solution ◽  
Positive Constant ◽  
Semilinear Elliptic Equation ◽  
Entire Solutions ◽  
Semilinear Elliptic

In this paper, we consider the nonhomogeneous semilinear elliptic equation,where λ ≥ 0, 1 < p < (N + 2)/(N − 2), if N ≥ 3, 1 < p < ∞, if N = 2, h(x) ∈ H−l(ℝN), 0 ≢ h(x) ≥ 0 in ℝN, K(x) is a positive, bounded and continuous function on ℝN. We prove that if K(x) ≥ K∞ > 0 in ℝN, and lim∣x∣⃗∞K(x) = K∞, then there exists a positive constant λ✶ such that (✶)λ has at least two solutions if λ ∈ (0, λ✶) and no solution if λ > λ✶. Furthermore, (✶)λ has a unique solution for λ = λ✶ provided that h(x) satisfies some suitable conditions. We also obtain some further properties and bifurcation results of the solutions of (1.1)λ at λ = λ✶.

Sign in / Sign up

Export Citation Format

Share Document