The existence of a positive solution to asymptotically linear scalar field equations

Source abstract

We consider the following elliptic equation: where m > 0, f ( x , u )/ u tends to a positive constant as u → + ∞. Here, the nonlinear term f ( x , u ) does not satisfy the usual condition, that is, for some θ > 0, which is important in using the mountain pass theorem. The aim of this paper is to discuss how to use the mountain pass theorem to show the existence of a positive solution to the present problem when we lose the above condition. Furthermore, if f ( x , u ) ≡ f ( u ), we also prove that the above problem has a ground state by using the artificial constraint method.