A new variational principle, convexity and supercritical Neumann problems

Abstract

Utilizing a new variational principle that allows dealing with problems beyond the usual locally compactness structure, we study problems with a supercritical nonlinearity of the type $ -\Delta u + u= a(x) f(u)$ in $ \Omega$ with $\partial_\nu u=0$ on $ \partial \Omega$. Here $\Omega$ is a bounded domain with certain symmetry assumptions. We find positive nontrivial solutions in the case of suitable supercritical nonlinearities $f$ by finding critical points of $I$ where \[ I(u)=\int_\Omega \left\{ a(x) F^* \left( \frac{-\Delta u + u}{a(x)} \right) - a(x) F(u) \right\} dx, \] over the closed convex cone $K_m$ of nonnegative, symmetric and monotonic functions in $H^1(\Omega)$ where $F'=f$ and where $ F^*$ is the Fenchel dual of $F$. We mention two important comments: firstly that 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 super-critical problems lacking the necessary compactness requirement. Secondly the energy $I$ is not at all related to the classical Euler-Lagrange energy associated with equation. After we have proven the existence of critical points $u$ of $I$ on $K_m$ we then unitize a new abstract variational approach (developed by one of the present authors in \cite{Mo,Mo2}) to show these critical points in fact satisfy $-\Delta u + u = a(x) f(u)$. 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.