Existence of Solutions to a Slightly Supercritical Pure Neumann Problem

.We show the existence and multiplicity of concentrating solutions to a pure Neumann slightly supercritical problem in a ball. This is the first existence result for this kind of problem in the supercritical regime. Since the solutions must satisfy a compatibility condition of zero average, all of them have to change sign. Our proofs are based on a Lyapunov–Schmidt reduction argument which incorporates the zero-average condition using suitable symmetries. Our approach also guarantees the existence and multiplicity of solutions to subcritical Neumann problems in annuli. More general symmetric domains (e.g., ellipsoids) are also discussed.Keywordssupercritical problemsLyapunov–Schmidt reductionsemilinear elliptic equationNeumann boundary conditionssymmetric solutionsMSC codes35B0735B3335B4435J91