Nonlinear scalar field $$(p_{1}, p_{2})$$ -Laplacian equations in $$\mathbb {R}^{N}$$ : existence and multiplicity

In this paper, we deal with the following class of \((p_{1}, p_{2})\)-Laplacian problems:

$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta _{p_{1}}u-\Delta _{p_{2}}u= g(u) \text{ in } \mathbb {R}^{N},\\ u\in W^{1, p_{1}}(\mathbb {R}^{N})\cap W^{1, p_{2}}(\mathbb {R}^{N}), \end{array} \right. \end{aligned}$$

where \(N\ge 2\), \(1<p_{1}<p_{2}\le N\), \(\Delta _{p_{i}}\) is the \(p_{i}\)-Laplacian operator, for \(i=1, 2\), and \(g:\mathbb {R}\rightarrow \mathbb {R}\) is a Berestycki-Lions type nonlinearity. Using appropriate variational arguments, we obtain the existence of a ground state solution. In particular, we provide three different approaches to deduce this result. Finally, we prove the existence of infinitely many radially symmetric solutions. Our results improve and complement those that have appeared in the literature for this class of problems. Furthermore, the arguments performed throughout the paper are rather flexible and can be also applied to study other p-Laplacian and \((p_1, p_2)\)-Laplacian equations with general nonlinearities.