Let \(K\) be a nonempty convex subset of a Hausdorff topological vector space \(X\) and let \(f:K\rightarrow X^*\) be a nonlinear operator. The paper deals with the study of the following class of variational inequalities: find \(x_0\in K\) such that \(\langle f(x_0),x-x_0\rangle\geq 0\), for all \(x\in K\). Problems of this type were originally studied in the 60's by Stampacchia. The author of this paper proves several existence results for the above class of densely pseudomonotone variational inequalities. There are also given some particular cases in reflexive Banach spaces which include previously known results. In another section of the paper, using recession directions, there are derived existence criteria for monotone and densely psudomonotone variational inequalities. Some of these results are applied in the last part of the paper to find generalized versions of the Browder-Minty theorem on the surjectivity of monotone operators.