This article aims to investigate the existence of bounded positive solutions of problem \( (P)\qquad \left\{ \begin{array}{ll} - \textrm{div} (a(x,u,\nabla u)) + A_t(x,u,\nabla u) = g(x,u) & \hbox {in }\Omega ,\\ u\ = \ 0 & \hbox {on }\partial \Omega , \end{array} \right. \) with \(A_t(x,t,\xi ) = \frac{\partial A}{\partial t}(x,t,\xi )\) , \(a(x,t,\xi ) = \nabla _\xi A(x,t,\xi )\) for a given \(A(x,t,\xi )\) which grows as \(|\xi |^p + |t|^p\) , \(p > 1\) , where \(\Omega \subseteq \mathbb {R}^N\) , \(N \ge 2\) , is an open connected domain with Lipschitz boundary and infinite Lebesgue measure, eventually \(\Omega = \mathbb {R}^N\) , which generalizes the modified Schrödinger equation \(\begin{aligned} - \text {div} ((A^*_1(x) + A^*_2(x)|u|^{s}) \nabla u)&+ \frac{s}{2} A^*_2(x)\ |u|^{s - 2} u\ |\nabla u|^2 + u\\&=\ |u|^{\mu -2}u \quad \text{ in } \mathbb {R}^3 \end{aligned}\) Under suitable assumptions on \(A(x,t,\xi )\) and g(x, t), problem (P) has a variational structure. Then, even in lack of radial symmetry hypotheses, one bounded positive solution of (P) can be found by passing to the limit on a sequence \((u_k)_k\) of bounded solutions on bounded domains. Furthermore, if stronger hypotheses are satisfied, either such a solution is nontrivial or a constant \(\bar{\lambda } > 0\) and a sequence of points \((y_k)_k \subset \mathbb {R}^N\) exist such that \( |y_k| \rightarrow +\infty \qquad \hbox {and}\qquad \int _{B_1(y_k)} |u_k|^p dx \ge \bar{\lambda }\quad \hbox {for all }k \ge 1, \) with \(B_1(y_k) = \{x \in \mathbb {R}^N: |x-y_k| < 1\}\) .
