We study a class of quasilinear elliptic problems with diffusion matrices that have at least one diagonal coefficient that blows up for a finite value of the unknown; the other coefficients being continuous with respect to the unknown (without any growth assumption). We introduce two equivalent notions of solutions for such problems and we prove an existence result in these frameworks. Under additional local assumptions on the coefficients, we also establish the uniqueness of the solution. In that case, and when the non-diagonal coefficients are bounded, this unique (generalized) solution is also the unique weak solution strictly less than the value where the diagonal coefficient blows up.