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 \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, $\Omega$ is a bounded domain in $\mathbb {R}^N$ which satisfies certain symmetry assumptions, $\Omega$ is a domain of “$m$ revolution" ($1\leq m<N$ 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 \[ I(u)=\int _\Omega \left \{ a(x) F^* \left ( \frac {-\Delta u + u}{a(x)} \right ) - a(x) F(u) \right \} dx \] 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: 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 $-\Delta u + u = a(x) f(u)$.

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.