Let \(\Omega\) be an unbounded sufficiently regular open set of \(\mathbb{R}^2\), \(n> 2\), and the linear differential uniformly elliptic operator \[ Lu= - \sum^n_{i, j= 1} a_{ij} u_{x_i x_j}+ \sum^n_{i= 1} a_i u_{x_i}+ au \] such that \(a_{ij}= a_{ji}\in L^\infty(\Omega)\), \(i, j= 1,\dots, n\). For this operator, the Dirichlet problem \(u\in W^2(\Omega)\cap W^1_0(\Omega)\) under some specific hypothesis on the coefficients has been studied previously by the same authors [Rend. Accad. Naz. Sci. XL, V. Ser., Mem. Mat. 18, No. 1, 41-56 (1994; Zbl 0833.35033)]. In the present paper, the same problem is studied, but under different conditions on the coefficients \(a_{ij}\).