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where β is an appropriate positive constant, see [1, 2, 4, 6, 7]. The question of uniqueness of ground states for (1.1) when f(u) has such exponential behavior has not previously been treated; the purpose of this paper is to make a beginning on this problem. Appropriate assumptions on the operator A will be given in the next section. On the other hand, for the nonlinearity f we consider the specific family of functions

Uniqueness of ground states for quasilinear elliptic equations in the exponential case