In this paper, we investigate the existence of a positive solution to the following p-Laplacian equation: \( -\Delta _p u + \lambda |u|^{p-2}u = f(u), \quad u \in W^{1,p}(\mathbb {R}^N), \quad N> p > 1, \) subject to the prescribed mass constraint \(\int _{\mathbb {R}^N} |u|^p \, \textrm{d}x = a > 0\) . Here, \(f \in C(\mathbb {R}, \mathbb {R})\) is a very general nonlinearity exhibiting Sobolev critical growth. Under suitable assumptions, we establish the existence of a mountain-pass normalized solution, which is also a ground state, for any given mass \(a > 0\) . By developing a novel min–max approach and establishing precise energy estimates, we overcome the lack of compactness and provide a unified solution to Soave’s open problems in the p-Laplacian setting.
