Singular set of solutions of semilinear elliptic equations

We study the behavior of the singular set {u=∇u=0} for solutions u to the semilinear elliptic equation Δu=f(x,u,∇u),x∈Ω, where Ω is an open set in Rn and |f(x,u,∇u)|≤A|u|p+B|∇u|q, where A,B≥0 and p,q∈(0,∞). We show that in dimension n=2 the singular set is a discrete set, and if min{p,q}...

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Bibliographic Details
Published in:Nonlinear analysis 2015-11, Vol.127, p.206-214
Main Author: Aghajani, Asadollah
Format: Article
Language:English
Subjects:
Density
Free boundary problems
Infinity
Mathematical analysis
Nonlinearity
Optimal growth
Optimization
Regularity
Singular set
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Summary:We study the behavior of the singular set {u=∇u=0} for solutions u to the semilinear elliptic equation Δu=f(x,u,∇u),x∈Ω, where Ω is an open set in Rn and |f(x,u,∇u)|≤A|u|p+B|∇u|q, where A,B≥0 and p,q∈(0,∞). We show that in dimension n=2 the singular set is a discrete set, and if min{p,q}
ISSN:0362-546X
1873-5215
DOI:10.1016/j.na.2015.07.005