Existence of solutions to supercritical Neumann problems via a new variational principle
| dc.contributor.author | Cowan, Craig | |
| dc.contributor.author | Moameni, Abbas | |
| dc.contributor.author | Salimi, Leila | |
| dc.date.accessioned | 2022-06-13T13:12:57Z | |
| dc.date.available | 2022-06-13T13:12:57Z | |
| dc.date.issued | 2017-09-13 | |
| dc.description.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. | |
| dc.description.department | Mathematics | |
| dc.format | Text | |
| dc.format.extent | 19 pages | |
| dc.format.medium | 1 file (.pdf) | |
| dc.identifier.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. | |
| dc.identifier.issn | 1072-6691 | |
| dc.identifier.uri | https://hdl.handle.net/10877/15907 | |
| dc.language.iso | en | |
| dc.publisher | Texas State University, Department of Mathematics | |
| dc.rights | Attribution 4.0 International | |
| dc.rights.uri | https://creativecommons.org/licenses/by/4.0/ | |
| dc.source | Electronic Journal of Differential Equations, 2017, San Marcos, Texas: Texas State University and University of North Texas. | |
| dc.subject | Variational principles | |
| dc.subject | Supercritical | |
| dc.subject | Neumann boundary condition | |
| dc.title | Existence of solutions to supercritical Neumann problems via a new variational principle | |
| dc.type | Article |