Existence and multiplicity of solutions to superlinear periodic parabolic problems
Date
2018-03-14
Authors
Godoy, Tomas
Kaufmann, Uriel
Journal Title
Journal ISSN
Volume Title
Publisher
Texas State University, Department of Mathematics
Abstract
Let Ω ⊂ ℝN be a smooth bounded domain and let α, b, c be three (possibly discontinuous and unbounded) T-periodic functions with c ≥ 0. We study existence and nonexistence of positive solutions for periodic parabolic problems Lu = λ(α(x, t)up - b(x, t)uq + c(x, t)) in Ω x ℝ with Dirichlet boundary condition, where λ > 0 is a real parameter and p > q ≥ 1. If α and b satisfy some additional conditions and p < (N + 2)/(N + 1) multiplicity results are also given. Qualitative properties of the solutions are discussed as well. Our approach relies on the sub and supersolution method (both to find the stable solution as well as the unstable one) combined with some facts about linear problems with indefinite weight. All results remain true for the corresponding elliptic problems. Moreover, in this case the growth restriction becomes p < N/(N - 1).
Description
Keywords
Periodic parabolic problems, Superlinear, Sub and supersolutions, Elliptic problems
Citation
Godoy, T., & Kaufmann, U. (2018). Existence and multiplicity of solutions to superlinear periodic parabolic problems. <i>Electronic Journal of Differential Equations, 2018</i>(70), pp. 1-12.
Rights
Attribution 4.0 International
Rights Holder
This work is licensed under a Creative Commons Attribution 4.0 International License.