q-Laplace equation involving the gradient on general bounded and exterior domains

The existence of positive singular solutions of

$$\begin{aligned} \left\{ \begin{array}{lcc} -\Delta _q u=(1+g(x))|\nabla u|^p &{}\quad \text {in}&{} B_1,\\ u=0&{}\quad \text {on}&{} \partial B_1, \end{array} \right. \end{aligned}$$(1)

is proved, where \(B_1\) is the unit ball in \({\mathbb {R}}^N\), \(N \ge 3\), \(2<q<N\), \(\frac{N(q-1)}{N-1}<p<q\) and \(g\ge 0\) is a Hölder continuous function with \(g(0) = 0\). Also, the existence of positive singular solutions of

$$\begin{aligned} \left\{ \begin{array}{lcc} -\Delta _q u=|\nabla u|^p &{}\quad \text {in}&{} \Omega ,\\ u=0&{}\quad \text {on}&{} \partial \Omega . \end{array} \right. \end{aligned}$$(2)

is proved, where \(\Omega \) is a bounded smooth domain in \({\mathbb {R}}^N\), \(N \ge 3\), \(2< q<N\) and \(\frac{N(q-1)}{N-1}<p<q\). Finally, the existence of a bounded positive classical solution of (2) with the additional property that \(\nabla u(x) \cdot x > 0\) for large |x| is proved, in the case of \(\Omega \) an exterior domain \({\mathbb {R}}^N\), \(N\ge 3\) and \(p >\frac{N(q-1)}{N-1}\).