The authors consider an obstacle problem of the form \begin{align*} \begin{cases} -\text{div}\ a(x,\nabla u)\in f(x,u,\nabla u)& \text{in }\Omega,\\ u\leq \Phi &\text{in }\Omega,\\ u=0&\text{on }\partial\Omega, \end{cases} \end{align*} where \(a\colon\overline{\Omega}\times\mathbb{R}^N\to\mathbb{R}^N\) is continuous and satisfies suitable structure conditions, \(\Phi\colon \Omega\to [0,+\infty]\) is a given obstacle and \(f\colon\Omega\times\mathbb{R}\times\mathbb{R}^N\to2^{\mathbb{R}}\) is a multivalued function depending on the gradient of the solution satisfying appropriate growth and coercivity properties. The main result in this paper shows that the solution set of the problem above is nonempty, bounded and closed. Its proof is based on the surjectivity theorem for multivalued functions given by the sum of a maximal monotone multivalued operator and a bounded multivalued pseudomonotone one.