On a zero-mass $(p,q)$-Laplacian equation involving subcritical and supercritical growth in $\mathbb R^N$

This paper is concerned with the zero-mass $(p,q)$-Laplacian equation $$ -\Delta_p u-\Delta_q u = |u|^{r-2}u+\lambda |u|^{s-2}u \quad \text{in } \mathbb{R}^N, $$ where $1<q<p<N$.

The exponents $r$ and $s$ may belong to either the subcritical or the supercritical range with respect to the critical Sobolev exponents $p^*$ and $q^*$.

We establish existence and nonexistence results and show that the sign of the parameter $\lambda$ determines the solvability regimes of the equation.

The existence proofs rely on variational methods, truncation arguments, and regularity theory, while the nonexistence results are derived from a suitable Pohozaev-type identity for the $(p,q)$-Laplacian operator.