The limiting behavior of solutions to \(p\)-Laplacian problems with convection and exponential terms (Q6654807)

In this paper, the authors consider the homogeneous Dirichlet problem for the equation \N\[\N-\Delta_p u=\lambda u^{q-1}+\beta u^{a-1}|\nabla u|^b+ m u^{l-1} e^{\alpha u^s},\N\]\Nwhere \(\Omega \subset \mathbb{R}^N\) is a smooth bounded domain, \(a, l \geq 1, b, s, \alpha>0\), and \(p>q \geq 1\). Under some assumptions of the parameters \(\lambda, \beta\) and \(m\), the authors show that the above problem has a positive solution. If, moreover, \(\lambda, \beta>0\) are arbitrarily fixed and \(m\) is sufficiently small, the problem has a positive solution \(u_p\), for all \(p\) sufficiently large, and \(u_p\) converges uniformly to the distance function to the boundary of \(\Omega\), as \(p \rightarrow \infty\).