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

Abstract

Abstract In this paper, we consider the following elliptic problem: − div g ( | ∇ g u | N − 2 ∇ g u ) + V ( x ) | u | N − 2 u = f ( x , u ) a ( x ) in M , ( P a ) $$ -\mathtt{div}_{g}\bigl( \vert \nabla_{g} u \vert ^{N-2}\nabla_{g} u \bigr)+V(x) \vert u \vert ^{N-2}u = \frac{f(x, u)}{a(x)}\quad \mbox{in } M, \qquad (P_{a}) $$ where ( M , g ) $(M, g)$ be a complete noncompact N-dimensional Riemannian manifold with negative curvature, N ≥ 2 $N\geq2$ , V is a continuous function satisfying V ( x ) ≥ V 0 > 0 $V(x) \geq V_{0 }> 0$ , a is a nonnegative function and f ( x , t ) $f(x, t)$ 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 $P_{a}$ ). Moreover, we also get a nontrivial weak solution to the perturbation problem.

Keywords

• Trudinger–Moser inequality
• Riemannian manifold
• Exponential growth
• The ground state solution