Existence of a positive solution for superlinear Laplacian equation via mountain pass theorem

Abstract

In this paper, we are going to show a nonlinear laplacian equation with the Dirichlet boundary value as follow has a positive solution: ( −∆u + V (x)u = g(x, u) x ∈ Ω u = 0 x ∈ ∂Ω where, ∆u = div(∇u) is the laplacian operator, Ω is a bounded domain in R³ with smooth boundary ∂Ω. At first, we show the equation has a nontrivial solution. next, using strong maximal principle, Cerami condition and a variation of the mountain pass theorem help us to prove critical point of functional I is a positive solution.