Nonexistence of positive solutions to a system of elliptic inequalities involving the Grushin operator

2021 ◽  
pp. 1-13
Author(s):  
Anh Tuan Duong ◽  
Quoc Hung Phan
Keyword(s):  
Positive Solutions ◽  
Grushin Operator ◽  
Elliptic Inequalities

Singular semilinear elliptic inequalities in the exterior of a compact set

2013 ◽  
Vol 143 (3) ◽  
pp. 563-588 ◽  
Author(s):  
Marius Ghergu ◽  
Steven D. Taliaferro
Keyword(s):  
Positive Solutions ◽  
Continuous Functions ◽  
Minimal Solution ◽  
Compact Set ◽  
Optimal Conditions ◽  
Isolated Singularities ◽  
Semilinear Elliptic ◽  
Elliptic Inequality ◽  
Isolated Point ◽  
Elliptic Inequalities

We study the semilinear elliptic inequality –Δu ≥ φ(δK (x))f(u) in ℝN / K, where φ, f are positive and non-increasing continuous functions. Here K ⊂ ℝN (N ≥ 3) is a compact set with finitely many components, each of which is either the closure of a C2 domain or an isolated point, and δK (x) = dist(x, ∂K). We obtain optimal conditions in terms of φ and f for the existence of C2-positive solutions. Under these conditions we prove the existence of a minimal solution and we investigate its behaviour around ∂K as well as the removability of the (possible) isolated singularities.

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