Morse indices and bifurcations of positive solutions of Δu + f(u) = 0 on ℝN

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Abstract

We consider the semilinear elliptic equation Δu + f(u) = 0, on ℝN, where f is of class C1 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 languageEnglish (US)
Pages (from-to)1407-1432
Number of pages26
JournalIndiana University Mathematics Journal
Volume50
Issue number3
StatePublished - Sep 1 2001

Keywords

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

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