## Abstract

We consider the semilinear elliptic equation Δu + f(u) = 0, on ℝ^{N}, where f is of class C^{1} and satisfies the conditions f(0) = 0, f′(0) < 0. By a ground state of this equation we mean a positive solution that decays to zero at infinity. Any such solution is necessarily radially symmetric about some point. If N = 1, the ground state, if it exists, is always unique (up to a shift in x), nondegenerate and has Morse index equal to one. We show that none of these statements is valid in general if N ≥ 2.

Original language | English (US) |
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Pages (from-to) | 1407-1432 |

Number of pages | 26 |

Journal | Indiana University Mathematics Journal |

Volume | 50 |

Issue number | 3 |

State | Published - Sep 1 2001 |

## Keywords

- Bifurcation
- Elliptic equations
- Ground states
- Morse index
- Nonuniqueness
- Positive solutions

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