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

We consider, for \(a,l\geq1,\) \(b,s,\alpha>0,\) and \(p>q\geq1,\) the homogeneous Dirichlet problem for the equation \(-\Delta_{p}u=\lambda u^{q-1}+\beta u^{a-1}\left\vert \nabla u\right\vert ^{b}+mu^{l-1}e^{\alpha u^{s}}\) in a smooth bounded domain \(\Omega\subset\mathbb{R}^{N}.\) We prove that under certain setting of the parameters \(\lambda,\) \(\beta\) and \(m\) the problem admits at least one positive solution. Using this result we prove that if \(\lambda,\beta>0\) are arbitrarily fixed and \(m\) is sufficiently small, then the problem has a positive solution \(u_{p},\) for all \(p\) sufficiently large. In addition, we show that \(u_{p}\) converges uniformly to the distance function to the boundary of \(\Omega,\) as \(p\rightarrow\infty.\) This convergence result is new for nonlinearities involving a convection term.