The limiting behavior of solutions to p-Laplacian problems with convection and exponential terms

We consider, for a, l1, b, s, > 0, and p> q1, the homogeneous Dirichlet problem for the equation -₏u= u^q-1+ u^a-1 u ^b+mu^l-1e^ u^{s} in a smooth bounded domain ^N. We prove that under certain setting of the parameters, and m the problem admits at least one positive solution. Using this result we prove that if, > 0 are arbitrarily fixed and m is sufficiently small, then the problem has a positive solution u₏, for all p sufficiently large. In addition, we show that u₏ converges uniformly to the distance function to the boundary of, as p. This convergence result is new for nonlinearities involving a convection term.