Existence of solutions to supercritical Neumann problems via a new variational principle




Cowan, Craig
Moameni, Abbas
Salimi, Leila

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Texas State University, Department of Mathematics


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.



Variational principles, Supercritical, Neumann boundary condition


Cowan, C., Moameni, A., & Salimi, L. (2017). Existence of solutions to supercritical Neumann problems via a new variational principle. <i>Electronic Journal of Differential Equations, 2017</i>(213), pp. 1-19.


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