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Journal of Dynamics and Differential Equations

, Volume 30, Issue 3, pp 1081–1118 | Cite as

Entire Solutions of Superlinear Problems with Indefinite Weights and Hardy Potentials

  • Matteo Franca
  • Andrea Sfecci
Article

Abstract

We provide the structure of regular/singular fast/slow decay radially symmetric solutions for a class of superlinear elliptic equations with an indefinite weight. In particular we are interested in the case where such a weight is positive in a ball and negative outside, or in the reversed situation. We extend the approach to elliptic equations in presence of Hardy potentials, i.e. to
$$\begin{aligned} \varDelta u +\frac{h(|\text {x}|)}{|\text {x}|^2} u+ f(u, |\text {x}|)=0 \end{aligned}$$
where h is not necessarily constant. By the use of Fowler transformation we study the corresponding dynamical systems, presenting the construction of invariant manifolds when the global existence of solutions is not ensured.

Keywords

Supercritical equations Hardy potentials Radial solutions Regular/singular ground states Fowler transformation Invariant manifold Continuability 

Mathematics Subject Classification

35J75 35J91 37D10 34C37 

Notes

Acknowledgements

The authors wish to thank the referee for the useful comments and suggestions which lead to an improvement of the paper.

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Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Dipartimento di Ingegneria Industriale e Scienze MatematicheUniversità Politecnica delle MarcheAnconaItaly

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