A generalization of a criterion for the existence of solutions to semilinear elliptic equations

Abstract

We prove an abstract result of existence of good generalized subsolutions for convex operators. Its application to semilinear elliptic equations leads to an extension of results by P.B-M.Pierre concerning a criterion for the existence of solutions to a semilinear elliptic or parabolic equation with a convex nonlinearity. We apply this result to the model problem \begin{document}$ -\Delta u = a |\nabla u|^p+ b|u|^q+f $\end{document} with Dirichlet boundary conditions where \begin{document}$ a,b>0 $\end{document} , \begin{document}$ p,q>1 $\end{document} . No other condition is made on \begin{document}$ p $\end{document} and \begin{document}$ q $\end{document} .