Journal of Inequalities and Applications (Mar 2020)
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.
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