Positive Solutions for a Singular Elliptic Equation Arising in a Theory of Thermal Explosion

In this paper, the thermal explosion model described by a nonlinear boundary value problem is studied. Firstly, we prove the comparison principle under nonlinear boundary conditions. Secondly, using the sub-super solution theorem, we prove the existence of a positive solution for the case K(x)>0, as well as the monotonicity of the maximal solution on parameter λ. Thirdly, the uniqueness of the solution for K(x)<0 is proved, as well as the monotonicity of the solutions on parameter λ. Finally, we obtain some new results for the existence of solutions, and the dependence on the λ for the case K(x) is sign-changing.