Positive Solution of Extremal Pucci’s Equations with Singular and Sublinear Nonlinearity

Positive Solution of Extremal Pucci’s Equations with Singular and Sublinear Nonlinearity In this paper, we establish the existence of a positive solution to $$\begin{aligned} \left\{ \begin{array}{ll} -\mathcal {M}^{+}_{\lambda ,\Lambda }(D^{2}u)=\frac{\mu k(x)f(u)}{u^{\alpha }}-\eta h(x)u^{q} &{}\quad \text {in }\;\Omega \\ u=0 &{}\quad \text {on }\;\partial \Omega , \end{array} \right. \end{aligned}$$ - M λ , Λ + ( D 2 u ) = μ k ( x ) f ( u ) u α - η h ( x ) u q in Ω u = 0 on ∂ Ω , where $$\Omega $$ Ω is a smooth bounded domain in $$\mathbb {R}^{n},~n\ge 1.$$ R n , n ≥ 1 . Under certain conditions on $$k,f~\text {and}~h,$$ k , f and h , using viscosity sub- and super solution method with the aid of comparison principle, we establish the existence of a unique positive viscosity solution. This work extends and complements the earlier works on semilinear and singular elliptic equations with sublinear nonlinearity. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Mediterranean Journal of Mathematics Springer Journals

Positive Solution of Extremal Pucci’s Equations with Singular and Sublinear Nonlinearity

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Publisher
Springer International Publishing
Copyright
Copyright © 2017 by Springer International Publishing AG
Subject
Mathematics; Mathematics, general
ISSN
1660-5446
eISSN
1660-5454
D.O.I.
10.1007/s00009-017-0950-6
Publisher site
See Article on Publisher Site

Abstract

In this paper, we establish the existence of a positive solution to $$\begin{aligned} \left\{ \begin{array}{ll} -\mathcal {M}^{+}_{\lambda ,\Lambda }(D^{2}u)=\frac{\mu k(x)f(u)}{u^{\alpha }}-\eta h(x)u^{q} &{}\quad \text {in }\;\Omega \\ u=0 &{}\quad \text {on }\;\partial \Omega , \end{array} \right. \end{aligned}$$ - M λ , Λ + ( D 2 u ) = μ k ( x ) f ( u ) u α - η h ( x ) u q in Ω u = 0 on ∂ Ω , where $$\Omega $$ Ω is a smooth bounded domain in $$\mathbb {R}^{n},~n\ge 1.$$ R n , n ≥ 1 . Under certain conditions on $$k,f~\text {and}~h,$$ k , f and h , using viscosity sub- and super solution method with the aid of comparison principle, we establish the existence of a unique positive viscosity solution. This work extends and complements the earlier works on semilinear and singular elliptic equations with sublinear nonlinearity.

Journal

Mediterranean Journal of MathematicsSpringer Journals

Published: Jun 12, 2017

References

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