A new variational principle, convexity, and supercritical Neumann problems

Abstract

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 (1) { − Δ u + u = a ( x ) f ( u ) a m p ; in Ω , u > 0 a m p ; in Ω , ∂ u ∂ ν = 0 a m p ; on ∂ Ω . \begin{equation}\tag {1} \begin {cases} -\Delta u + u = a(x) f(u) & \text {in $\Omega $}, \\ u>0 & \text {in $\Omega $}, \\ \frac {\partial u}{\partial \nu } = 0 & \text {on $\partial \Omega $}. \end{cases} \end{equation} To be more precise, Ω \Omega is a bounded domain in R N \mathbb {R}^N which satisfies certain symmetry assumptions, Ω \Omega is a domain of “ m m revolution" ( 1 ≤ m > N 1\leq m>N and the case of m = 1 m=1 corresponds to radial domains), and a > 0 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 f by finding critical points of I I where \[ I ( u ) = ∫ Ω { a ( x ) F ∗ ( − Δ u + u a ( x ) ) − a ( x ) F ( u ) } d x I(u)=\int _\Omega \left \{ a(x) F^* \left ( \frac {-\Delta u + u}{a(x)} \right ) - a(x) F(u) \right \} dx \] over the closed convex cone K m K_m of nonnegative, symmetric, and monotonic functions in H 1 ( Ω ) H^1(\Omega ) where F ′ = f F’=f and where F ∗ F^* is the Fenchel dual of F F . We mention two important comments: First, there is a hidden symmetry in the functional I 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 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 u of I I on K m K_m , we then unitize a new abstract variational approach to show that these critical points in fact satisfy − Δ u + u = a ( x ) f ( u ) -\Delta u + u = a(x) f(u) . In the particular case of f ( u ) = | u | p − 2 u f(u)=|u|^{p-2} u we show the existence of positive nontrivial solutions beyond the usual Sobolev critical exponent.