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

In this paper, we consider the following problem {-Delta u = lambda(|u|(r-1)u + a|del u|(s)) + f(x, u) in Omega, u = 0 on partial derivative Omega, (0.1) where lambda > 0, r is an element of(0, 1), s is an element of(0, 2), a is an element of R and f is an element of C(Omega x R). The term f can be exponential growth at infinity. 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 lambda is an element of(0, lambda*) with lambda* > 0. Particularly, problem (0.1) admits a positive solution and a negative solution in the case a = 0 for lambda is an element of(0, lambda*). (c) 2022 Elsevier Inc. All rights reserved.