Electron. J. Differential Equations, Vol. 2018 (2018), No. 70, pp. 1-12.

Existence and multiplicity of solutions to superlinear periodic parabolic problems

Tomas Godoy, Uriel Kaufmann

Abstract:
Let $\Omega\subset\mathbb{R}^{N}$ be a smooth bounded domain and let a,b,c be three (possibly discontinuous and unbounded) T-periodic functions with $c\geq0$. We study existence and nonexistence of positive solutions for periodic parabolic problems $Lu=\lambda(a(x,t)u^p-b(x,t) u^{q}+c(x,t))$ in $\Omega\times\mathbb{R}$ with Dirichlet boundary condition, where $\lambda>0$ is a real parameter and $p>q\geq 1$. If a 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 $p<N/(N-1)$.

Submitted November 3, 2017. Published March 14, 2018.
Math Subject Classifications: 35K20, 35K60, 35B10.
Key Words: Periodic parabolic problems; superlinear; sub and supersolutions; elliptic problems.

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Tomas Godoy
FaMAF, Universidad Nacional de Córdoba
(5000) Córdoba, Argentina
email: godoy@mate.uncor.edu
Uriel Kaufmann
FaMAF, Universidad Nacional de Córdoba
(5000) Córdoba, Argentina
email: kaufmann@mate.uncor.edu

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