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The paper concerns the existence of weak solutions to nonlinear elliptic equations of the form A(u) + g(x,u,∇u) = f, where A is an operator from an appropriate anisotropic function space to its dual and the right hand side term is in $L^{1+m}$ with 0 < m < 1. We assume a sign condition on the nonlinear term g, but no growth restrictions on u.

Via critical point theory we establish the existence and regularity of solutions for the quasilinear elliptic problem ⎧ $div(x,\nabla u)+{a\left(x\right)\left|u\right|}^{p-2}u={g\left(x\right)\left|u\right|}^{p-2}u+h\left(x\right){\left|u\right|}^{s-1}u$ in ${ℝ}^N$ ⎨ ⎩ u > 0, $li{m}_{\left|x\right|\to \infty }u\left(x\right)=0$, where 1 < p < N; a(x) is assumed to satisfy a coercivity condition; h(x) and g(x) are not necessarily bounded but satisfy some integrability restrictions.