We use a new variational principle to obtain a positive solution of (Formula Presented) in B1, with Neumann boundary conditions where B11 is the unit ball in ℝN, a in nonnegative, radial and increasing and p > 2. Note that for N ≥ 3 this includes supercritical values of p. We find critical points of the functional (formula present) over the set of {u Є H1 rad rad(B1): 0 ≤ u, u is increasing}, where q is the conjugate of p. We would like to emphasize the energy functional I is different from the standard Euler-Lagrange functional associated with the above equation, i.e. (Formula Presented) The novelty of using I instead of E is the hidden symmetry in I generated by 1/pƒB1 a(|x|)|u|p dx and its Fenchel dual. Additionally we are able to prove the existence of a positive nonconstant solution, in the case a(|x|) = 1, relatively easy and without needing to cut off the supercritical nonlinearity. Finally, we use this new approach to prove existence results for gradient systems with supercritical nonlinearities.

Additional Metadata
Keywords Neumann boundary conditions, Supercritical, Variational principles
Journal Electronic Journal of Differential Equations
Citation
Cowan, C. (Craig), Momeni, A, & Salimi, L. (Leila). (2017). Existence of solutions to supercritical Neumann problems via a new variational principle. Electronic Journal of Differential Equations, 2017.