Existence of solutions to supercritical Neumann problems via a new variational principle
Date
2017-09-13
Authors
Cowan, Craig
Moameni, Abbas
Salimi, Leila
Journal Title
Journal ISSN
Volume Title
Publisher
Texas State University, Department of Mathematics
Abstract
We use a new variational principle to obtain a positive solution of
-∆u + u = α(|x|)|u|p-2u in B1,
with Neumann boundary conditions where B1 is the unit ball in ℝN, α in nonnegative, radial and increasing and p > 2. Note that for N ≥ 3 this includes supercritical values of p. We find critical points of the functional
I(u) ≔ 1/q ∫B1 α(|x|)1-q| -∆u + u|q dx - 1/p ∫B1 α(|x|)|u|p dx,
over the set of {u ∈ H1rad (B1) : 0 ≤ u, u is increasing}, where q is the conjugate of p. We would like to emphasize the energy functional I is different from the standard Euler-Lagrange functional associated with the above equation, i.e.
E(u) ≔ ∫B1 |∇u|2 + u2/2 dx - ∫B1 α(|x|)|u|p/p dx.
The novelty of using I instead of E is the hidden symmetry in I generated by 1/p ∫B1 α(|x|)|u|p dx and its Fenchel dual. Additionally we were able to prove the existence of a positive nonconstant solution, in the case α(|x|) = 1, relatively easy and without needing to cut off the supercritical nonlinearity. Finally, we use this new approach to prove existence results for gradient systems with supercritical nonlinearities.
Description
Keywords
Variational principles, Supercritical, Neumann boundary condition
Citation
Cowan, C., Moameni, A., & Salimi, L. (2017). Existence of solutions to supercritical Neumann problems via a new variational principle. Electronic Journal of Differential Equations, 2017(213), pp. 1-19.
Rights
Attribution 4.0 International