Existence results for a quasilinear elliptic problem with a gradient term via shooting method

Covei, Dragoş-Pătru

doi:10.1016/j.amc.2011.09.047

Cited by 2 publications

(1 citation statement)

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“…The function f : R → R represents the nonlinearity and b : M → R is a regularizing coefficient. The semi-linear problem (1) has been studied in several configuration manifolds like R n , S n or some surfaces of revolution; see [14,2,1,4,3] among others. On these cases, the existence of solutions have been obtained with an analytical approach after a change of variables.…”

Section: Introductionmentioning

confidence: 99%

On polar actions invariant solutions of semi-linear equations on manifolds

Becerra

Galvis

Martínez

2019

Journal of Mathematical Analysis and Applications

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In this paper we put together some tools from differential topology and analysis to study second order semi-linear partial differential equations on a Riemannian manifold M . We look for solutions that are constants along orbits of a given group action. Using some results obtained by Helgason in [J DIFFER GEOM,6(3), 411-419] we are able to write a (reduced) second order semi-linear problem on a submanifold Σ. This submanifold is, in certain sense, transversal to the orbits of the group actions and its existence is assumed. We describe precise conditions on the Riemannian Manifold M and the submanifold Σ in order to be able to write the reduced equation on Σ. These conditions are satisfied by several particular cases including some examples treated separately in the literature such as the sphere, surfaces of revolution and others. Our framework also includes the setup of polar actions or exponential coordinates. Using this procedure, we are left with a second order semi-linear equation posed on a submanifold. In particular, if the submanifold Σ is one-dimensional, we can use suitable tools from analysis to obtain existence and properties of solutions.

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