We deal with (r, s)-linear Weingarten two-sided hypersurfaces immersed in the \((n+1)\) -dimensional real projective space \(\mathbb{R}\mathbb{P}^{n+1}\) , namely, two-sided hypersurfaces of \(\mathbb{R}\mathbb{P}^{n+1}\) whose higher order mean curvatures \(H_{r+1}\) and \(H_{s+1}\) are linearly related, with \(0\le r\le s\le n-1\) . Under suitable restrictions on the geometry of these hypersurfaces, we prove the nonexistence of strongly stable (r, s)-linear Weingarten closed two-sided hypersurfaces immersed in a certain region determined by a geodesic sphere of \(\mathbb{R}\mathbb{P}^{n+1}\) . We also obtain a uniqueness result for strongly stable (r, s)-linear Weingarten closed two-sided hypersurfaces of \(\mathbb{R}\mathbb{P}^{n+1}\) .
