Multiplicity of positive solution for an indefinite elliptic equation on R2R2 with exponential nonlinearities

In this paper, we consider the multiplicity of positive solution to the equation−Δu=λu+h(x)upeu,x∈R2, with h(x)h(x) a sign-changing function, p>1p>1 a constant and λ a parameter. We first use a moving plane argument to get a priori bounds for the positive solutions of this equation. Then we obtain multiple positive solutions through a squeezing method, which overcomes the lack of compactness of the problem.