| Abstract: |
| We focus on positive radial solutions to the Dirichlet problem associated with the p-Laplace eigenvalue scalar curvature equation.
The scalar curvature function K is assumed to be smooth, positive and to satisfy the l-flatness condition in a neighborhood of zero.
We show that the number of radial positive solutions undergoes a bifurcation phenomenon: if K is steep enough at 0, the problem admits one solution for every positive eigenvalue $\lambda$, while if K is too flat at 0, the problem admits no solutions for $\lambda$ small and two solutions for $\lambda$ large.
The existence of the second solution, ensured by an additional monotonicity assumption on K, is new, even in the classical Laplace case.
The proofs are based on Fowler transformation, invariant manifold theory and phase plane analysis. |
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