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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



A new variational principle, convexity, and supercritical Neumann problems

Authors: Craig Cowan and Abbas Moameni
Journal: Trans. Amer. Math. Soc. 371 (2019), 5993-6023
MSC (2010): Primary 35J15; Secondary 58E30
Published electronically: February 1, 2019
MathSciNet review: 3937316
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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

$\displaystyle \left \{\begin {array}{ll} -\Delta u + u= a(x) f(u) & \mbox { in ... ...ial u}{\partial \nu }= 0 & \mbox { on } {\partial \Omega }. \end{array}\right .$ (1)

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

$\displaystyle 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)=\vert u\vert^{p-2} u$ we show the existence of positive nontrivial solutions beyond the usual Sobolev critical exponent.

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Additional Information

Craig Cowan
Affiliation: University of Manitoba, Winnipeg, Manitoba, Canada

Abbas Moameni
Affiliation: School of Mathematics and Statistics, Carleton University, Ottawa, Ontario, Canada

Keywords: Variational principles, supercritical, Neumann BC
Received by editor(s): April 11, 2016
Received by editor(s) in revised form: June 7, 2016, December 7, 2016, and March 13, 2017
Published electronically: February 1, 2019
Additional Notes: Both authors are pleased to acknowledge the support of the National Sciences and Engineering Research Council of Canada.
Article copyright: © Copyright 2019 American Mathematical Society