Utilizing a new variational principle that allows us to deal with problems beyond the usual locally compact structure, we study problems with a supercritical nonlinearity of the type To be more precise, O is a bounded domain in RN which satisfies certain symmetry assumptions, O is a domain of “m revolution” (1 ≤ m < N and the case of m = 1 corresponds to radial domains), and a > 0 satisfies compatible symmetry assumptions along with monotonicity conditions. We find positive nontrivial solutions of (1) in the case of suitable supercritical nonlinearities f by finding critical points of I where over the closed convex cone K m of nonnegative, symmetric, and monotonic functions in H1(ω) where F’ = f and where Fz.ast; is the Fenchel dual of F. We mention two important comments: First, there is a hidden symmetry in the functional I due to the presence of a convex function and its Fenchel dual that makes it ideal to deal with supercritical problems lacking the necessary compactness requirement. Second, the energy I is not at all related to the classical Euler-Lagrange energy associated with (1). After we have proven the existence of critical points u of I on K m , we then unitize a new abstract variational approach to show that these critical points in fact satisfy -δu+u = a(x)f(u). In the particular case of f(u) = |u| p-2 u we show the existence of positive nontrivial solutions beyond the usual Sobolev critical exponent.

Neumann BC, Supercritical, Variational principles
Transactions of the American Mathematical Society
School of Mathematics and Statistics

Cowan, C. (Craig), & Momeni, A. (2019). A new variational principle, convexity, and supercritical neumann problems. Transactions of the American Mathematical Society, 371(9), 5993–6023. doi:10.1090/tran/7250