convex–concave problem with a nonlinear boundary condition

We study the asymptotic behavior of ground states of quasilinear elliptic problems with two vanishing parameters. Thanks to an additional (fixed) parameter, we show that two different critical exponents play a crucial role in the asymptotic analysis, giving an explanation of the phenomena discovered in Gazzola et al. (Asymptotic behavior of ground states of quasilinear elliptic problems with two vanishing parameters, Ann. Inst. H. Poincare Anal. Non Lineaire, to appear) and Gazzola and Serrin (Ann. Inst. H. Poincare Anal. Non Lineaire 19 (2002) 477).