Multiple solutions for elliptic equations with exponential nonlinearity term combined with convection term in dimension two

Abstract

In this paper, we consider the following problem(0.1){−Δu=λ(|u|r−1u+a|∇u|s)+f(x,u)inΩ,u=0on∂Ω, where λ>0, r∈(0,1), s∈(0,2), a∈R and f∈C(Ω×R). The term f can be exponential growth at ∞. Convection term, namely gradient term, makes the problem (0.1) invariational. Under suitable conditions imposed on f, through the approximation scheme we prove that problem (0.1) admits a positive solution if a⩾0 and a negative solution if a⩽0 for λ∈(0,λ⁎) with λ⁎>0. Particularly, problem (0.1) admits a positive solution and a negative solution in the case a=0 for λ∈(0,λ⁎).