Elliptic variational inequalities of the form \[ \langle Au, u- v\rangle+ j(u)\leq j(v)\qquad \forall v\in V\tag{1} \] have been widely considered in the literature in the coercive case, that is when the mapping \(u\mapsto \langle Au, u\rangle+ j(u)\) has a superlinear growth as \(\| u\|\to +\infty\). Here \(V\) is a reflexive separable Banach space, \(j: V\to \overline{{\mathbf R}}\) is a proper convex l.s.c. functional, and \(A: V\to V'\) is a given operator. In this case, under the so-called pseudomonotonicity assumption on \(A\), existence results are well-known for problem (1). On the other side, when the coerciveness assumption is dropped, in order to have the existence of a solution for (1), compatibility conditions between \(A\) and \(j\) must be added. The case when \(A\) is the gradient of a functional has been extensively studied by \textit{C. Baiocchi}, the reviewer, \textit{F. Gastaldi} and the author [Arch. Ration. Mech. Anal. 100, No. 2, 149-189 (1988; Zbl 0646.73011)] who developed a general theory for noncoercive minimization problems; on the other hand, the case when \(A\) is linear and positively semidefinite has been considered by \textit{C. Baiocchi}, \textit{F. Gastaldi} and the author [Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 13, 617- 659 (1986; Zbl 0644.49004)]. In this paper the author considers the case when \(A\) is pseudomonotone and noncoercive, and obtains necessary conditions, as well as sufficient conditions for existence. The results are then applied to the obstacle problem with a convection term (Section 4), to the stationary Navier- Stokes system (Section 5), and to the more abstract problems of closure for the sum of convex sets (Section 6) and of lower semicontinuity for the inf-convolution of convex functions.