Abstract
We consider here solutions of a nonlinear Neumann elliptic equation Δu + f (x, u) = 0 in Ω, ∂u/∂ν = 0 on ∂Ω, where Ω is a bounded open smooth domain in \({\mathbb{R}^N, N\geq2}\) and f satisfies super-linear and subcritical growth conditions. We prove that L ∞−bounds on solutions are equivalent to bounds on their Morse indices.
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Harrabi, A., Ahmedou, M.O., Rebhi, S. et al. A priori estimates for superlinear and subcritical elliptic equations: the Neumann boundary condition case. manuscripta math. 137, 525–544 (2012). https://doi.org/10.1007/s00229-011-0488-z
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DOI: https://doi.org/10.1007/s00229-011-0488-z