The main goal in this paper is to prove the existence of radial positive solutions of the quasilinear elliptic systemwhere Ω is a ball in R N and f , g are positive continuous functions satisfying f (x, 0, 0) = g(x, 0, 0) = 0 and some growth conditions which correspond, roughly speaking, to superlinear problems. Two different sets of conditions, called strongly and weakly coupled, are given in order to obtain existence. We use the topological degree theory combined with the blow up method of Gidas and Spruck. When Ω = R N , we give some sufficient conditions of nonexistence of radial positive solutions for Liouville systems.
We study the existence of bounded solutions to the elliptic systemThe method used is a shooting technique. We are concerned with the existence of a negative subsolution and a nonnegative supersolution in the sense of Hernandez; then we construct some compact operator T and some invariant set K where we can use the Leray Schauder's theorem.
The aim of this work is to establish the existence of infinitely many solutions to gradient elliptic system problem, placing only conditions on a potential function H, associated to the problem, which is assumed to have an oscillatory behaviour at infinity. The method used in this paper is a shooting technique combined with an elementary variational argument. We are concerned with the existence of upper and lower solutions in the sense of Hernández.
In this paper, we discuss the solvability of the nonlinear parabolic systems associated to the nonlinear parabolic equation: for $i=1,2$\begin{equation*}\mathcal{(S)} \left\{\begin{array}{lll}\displaystyle\frac{\partial u_{i}}{\partial t} -div(a(x,t,u_{i},\nabla u_{i}))+ g_{i}(x,t,u_{i},\nabla u_i)) =f_{i}(x,u_{1},u_{2})\, \quad & \mbox{in}\quad & Q_{T},\\\displaystyle u_{i}(x,t)=0 &\mbox{on}& \partial\Omega \times(0,T),\\\displaystyle u_{i}(x,(t=0))=u_{i,0}(x) & \mbox{in} & \Omega,\end{array}%\right.\end{equation*}with the source $f$ is merely integrable. The operator $\displaystyle A(u)= div \Big (a(x,t,u_{i},\nabla u_{i})\Big)$is a generalized Leray-Lions operator defined on the inhomogeneous Musielak-Orlicz spaces (the vector field $\displaystyle a(x,t,u_i,\nabla u_i)$ have a growth prescribed by a generalized N-function). The non linearity $g_{i}$ is a Carath\'{e}odory function satisfy the some condition.
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