On the study of solutions for a non linear differential equation on compact Riemannian Manifolds
Abstract
In this paper we study the existence of solutions for a class of non-linear differential equation on compact Riemannian manifolds. We establish a lower and upper solutions' method to show the existence of a smooth positive solution for the equation (EQ1) \begin{equation} \label{E4} \Delta u \ + \ a(x)u \ = \ f(x)F(u) \ + \ h(x)H(u), (EQ1) \end{equation} where \ $a, \ f, \ h$ \ are positive smooth functions on $M^n$, a $n-$dimensional compact Riemannian manifold, and \ $ F, \ H$ \ are non-decreasing smooth functions on $\mathbb{R}$. In \cite{djadli} the equation (EQ1) was studied when $F(u)=u^{2^{\ast}-1} $ and $H(u)=u^q$ in the Riemannian context, i.e., \begin{equation} \label{E3} \Delta u \ + \ a(x)u \ = \ f(x)u^{2^{\ast}-1} \ + \ h(x)u^q, (EQ2) \end{equation} \nd where \ $0 \ < \ q \ < 1$. In \cite{correa} Corrêa, Gonçalves and Melo studied an equation of the type equation (EQ2), in the Euclidean context.
- Publication:
-
arXiv e-prints
- Pub Date:
- November 2016
- DOI:
- 10.48550/arXiv.1611.02214
- arXiv:
- arXiv:1611.02214
- Bibcode:
- 2016arXiv161102214S
- Keywords:
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- Mathematics - Differential Geometry;
- Primary 53C21;
- Secondary 35J60