Existence and multiplicity of solutions to superlinear periodic parabolic problems

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 $p>q\geq1$. 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 $p<N/(N-1)$.

Keywords

• Periodic parabolic problems
• superlinear
• sub and supersolutions
• elliptic problems