Abstract. A nonnegative nontrivial solution of the quasilinear elliptic di¤erential equation (1.1) below on the entire space is obtained. The function fðtÞt in the principal part is nonhomogeneous and bðtÞt has the critical Orlicz-Sobolev growth with respect to f.

We study a quasilinear elliptic problemat t = 0 and ∞, the existence of multiple positive solutions is proved by using the variational arguments in the Orlicz-Sobolev spaces.

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A variational problem for a functional with slowly growing principal part and involving critical Orlicz–Sobolev lower term with respect to the principal part is discussed. The principal part of the functional is not Fréchet differentiable. The lack of differentiability and the critical growth rate of the lower term demand a precise compactness argument in the variational approach. A non-negative solution for the Euler equation is given.

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