Resumen de Periodic parabolic problems with nonlinearities indefinite in sign

T. Godoy, U. Kaufmann

• Let $\Omega\subset\mathbb{R}^{N}$ be a smooth bounded domain. We give sufficient conditions (which are also necessary in many cases) on two nonnegative functions $a$, $b$ that are possibly discontinuous and unbounded for the existence of nonnegative solutions for semilinear Dirichlet periodic parabolic problems of the form $Lu=\lambda a\left( x,t\right) u^{p}-b\left( x,t\right) u^{q}$ in $\Omega\times\mathbb{R}$, where $0 < p, q < 1$ and $\lambda > 0 $. In some cases we also show the existence of solutions $u_{\lambda}$ in the interior of the positive cone and that $u_{\lambda}$ can be chosen such that $\lambda\rightarrow u_{\lambda}$ is differentiable and increasing. A uniqueness theorem is also given in the case $p\leq q$. All results remain valid for the corresponding elliptic problems.