The authors establish existence results of bounded positive and radially symmetric weak solutions, with locally Hölderian gradient, to the quasilinear elliptic problem \[ \begin{cases} -\Delta_p u-\beta \Delta_q u=g(u),&\text{ in }\mathbb{R}^N,\\ u(x)\rightarrow 0 &\text{ as }|x|\rightarrow +\infty, \end{cases} \] where \(N\geq 3\), \(p\in (1,N)\), \(q\in (p,+\infty)\), \(\beta \in (0,+\infty)\) and \(g:\mathbb{R}\rightarrow \mathbb{R}\) is a continuous function which is identically zero in \((-\infty,0)\) and satisfies
(a) \(\sup_{\zeta\in (0,+\infty)}\int_0^\zeta g(s)ds>0\),
(b) \(\limsup_{s\rightarrow +\infty}\frac{g(s)}{s^{q^*-1}}\leq 0\),
(c) either \(\limsup_{s\rightarrow0^+}\frac{g(s)}{s^{l-1}}\leq 0\), for all \(l\in [p,p^*]\),
or \(-\infty<\liminf_{s\rightarrow0^+}\frac{g(s)}{s^{l-1}}\leq \limsup_{s\rightarrow0^+}\frac{g(s)}{s^{l-1}}<0\), for some \(l\in [p,p^*)\),
where \(p^*=\frac{pN}{N-p}\), \(q^*=\frac{qN}{N-q}\), if \(q<N\), and \(q^*\in (\max\{q,p^*\},+\infty)\), if \(q\geq N\).

The proof is based on the Mountain Pass Theorem and on the well-known Jeanjean’s monotonicity trick. Under the same assumptions, the authors show that there exists a radially symmetric solution which minimizes the associated energy functional over the set of all radially symmetric solutions.
Using similar techniques, the authors also extend the above results to the \(k\)-order approximation of the Born-Infeld equation \[ -\operatorname{div}\Biggl(\frac{\nabla u}{\sqrt{1-b^{-2}|\nabla u|^2}}\Biggr)=g(u), \text{ in }\mathbb{R}^N. \]