The existence of ground state solution to elliptic equation with exponential growth on complete noncompact Riemannian manifold

摘要

In this paper, we consider the following elliptic problem: $$ -mathtt{div}_{g}igl( ert abla_{g} u ert ^{N-2}abla_{g} u igr)+V(x) ert u ert ^{N-2}u = rac{f(x, u)}{a(x)}quad mbox{in } M, qquad (P_{a}) $$ where $(M, g)$ be a complete noncompact N-dimensional Riemannian manifold with negative curvature, $Ngeq2$, V is a continuous function satisfying $V(x) geq V_{0 } 0$, a is a nonnegative function and $f(x, t)$ has exponential growth with t in view of the Trudinger–Moser inequality. By proving some estimates together with the variational techniques, we get a ground state solution of ($P_{a}$). Moreover, we also get a nontrivial weak solution to the perturbation problem.