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.

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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