In this paper we discuss the nonexistence of stable and stable at infinity weak solutions to the following \(p-\) polyharmonic problem \(\begin{aligned} \Delta _{p}^r u =c_1u_+^{q_1}-c_2u_-^{q_2} \text{ in } \; {\mathbb {R}}^n, \end{aligned}\) where \(\Delta ^r_p\) is the \(p-\) polyharmonic operator, \(p\ge 2,\) \(c_1, \;c_2>0\) and \(q_1, \,q_2\) are subcritical. To provide the basic energy estimate we employ a new interpolation inequality to control the integral \(\displaystyle \int _{{\mathbb {R}}^n} |\nabla ^q v|^p|\nabla ^ {r-q}\phi |^p\) by the \(L^p\) -norm of \(\phi \nabla ^r v\) (respectively \(\nabla ^r( v\phi )\) ) and \(v\nabla ^r\phi ,\) where \(v\in W^{r,p}_{loc}({\mathbb {R}}^n)\) \(\phi \in C^\infty _c({\mathbb {R}}^n)\) a cut-off function and \(1\le q\le r-1\) . Our results improve and extend previous Liouville theorems stated in Harrabi (Annali di Matematica Pura ed Applicata. 198, 1675–1692 (2019)), Harrabi and Mtiri (Liouville-type theorems and existence results for stable at infinity solutions of higher order m-polyharmonic problems, To appear in JMAA (2021)), Mtiri and Ye (Nonlinearity 32, 910–926 (2019)). Particularly, we remove the exponential growth condition imposed on unbounded solutions in Harrabi (Annali di Matematica Pura ed Applicata. 198, 1675–1692 (2019)), Harrabi and Mtiri (Liouville-type theorems and existence results for stable at infinity solutions of higher order m-polyharmonic problems, To appear in JMAA (2021)).
