The Gel'fand Problem for the Biharmonic Operator
Résumé
We study stable and finite Morse index solutions of the equation Δ2 = eu. If the equation is posed in ℝN, we classify radial stable solutions. We then construct nonradial stable solutions and we prove that, unlike the corresponding second order problem, no Liouville-type theorem holds, unless additional information is available on the asymptotics of solutions at infinity. Thanks to this analysis, we prove that stable solutions of the equation on a smoothly bounded domain (supplemented with Navier boundary conditions) are smooth if and only if N ≦ 12. We find an upper bound for the Hausdorff dimension of their singular set in higher dimensions and conclude with an a priori estimate for solutions of bounded Morse index, provided they are controlled in a suitable Morrey norm. © 2013 Springer-Verlag Berlin Heidelberg.