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: $$ -\mathtt{div}_{g}\bigl( \vert \nabla_{g} u \vert ^{N-2}\nabla_{g} u \bigr)+V(x) \vert u \vert ^{N-2}u = \frac{f(x, u)}{a(x)}\quad \mbox{in } M, \qquad (P_{a}) $$ where $(M, g)$ be a complete noncompact N-dimensional Riemannian manifold with negative curvature, $N\geq2$, 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.

[学科分类] 电力

[关键词] Trudinger–Moser inequality;Riemannian manifold;Exponential growth;The ground state solution