On the structure of positive solutions to an elliptic problem arising in thin film equations

We consider the structure of positive radial solutions ofΔu+u−α−u−β=0in BR,u=0on ∂BR, where BRBR is a ball in RNRN with radius R  . When 0<α<β<10<α<β<1, we show that there exists R∗>0R∗>0 such that when R>R∗R>R∗, the Dirichlet problem has exactly two radial solutions; when R=R∗R=R∗, the solution is unique and there is no solution for R0R>0, the radial solution exists and is unique.