Leray–Lions operators with logarithmic growth

In this paper we prove existence, uniqueness, and regularity results for solutions of nonlinear elliptic equations in which the differential operator has logarithmic growth with respect to the gradient. The solution will belong to the Sobolev space W01,1(Ω), to the Orlicz–Sobolev space generated by the function tlog⁡(1+|t|)tlog⁡(1+|t|), but not to any Sobolev space W01,p(Ω), with p>1p>1.