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

Abstract

This paper is concerned with the zero-mass $(p,q)$-Laplacian equation $$-{\Delta }_{p}u-{\Delta }_{q}u=|u{|}^{r-2}u+\lambda |u{|}^{s-2}u\text{in }{\mathbb{R}}^N,$$ 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 $p^{\ast }$ and $q^{\ast }$. 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.

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