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 \[ I(u)=\int_Ω\left\{ a(x) F^* \left( \frac{-Δu + u}{a(x)} \right) - a(x) F(u) \right\} dx, \] over the closed convex cone $K_m$ of nonnegative, symmetric and monotonic functions in $H^1(\Omega )$ where $F^{\prime }=f$ and where $F^{\ast }$ 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.

References in corpus (3)

• Increasing radial solutions for Neumann problems without growth restrictions
• A note on the radial solutions for the supercritical Henon equation
• Multiple positive solutions of the stationary Keller-Segel system