# Entire Solutions of Superlinear Problems with Indefinite Weights and Hardy Potentials

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## 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 where

$$\begin{aligned} \varDelta u +\frac{h(|\text {x}|)}{|\text {x}|^2} u+ f(u, |\text {x}|)=0 \end{aligned}$$

*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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