  For $p\geq 1$ and $(g_{ij})_{1\leq i,j\leq n}$ being a matrix of i.i.d.
standard Gaussian entries, we study the $n$-limit of the
$\ell_p$-Gaussian-Grothendieck problem defined as
\begin{align*}\max\Bigl\{\sum_{i,j=1}^n g_{ij}x_ix_j: x\in
\mathbb{R}^n,\sum_{i=1}^n |x_i|^p=1\Bigr\}.\end{align*} The case $p=2$
corresponds to the top eigenvalue of the Gaussian Orthogonal Ensemble; when
$p=\infty$, the maximum value is essentially the ground state energy of the
Sherrington-Kirkpatrick mean-field spin glass model and its limit can be
expressed by the famous Parisi formula. In the present work, we focus on the
cases $1\leq p<2$ and $2<p<\infty.$ For the former, we compute the limit of the
$\ell_p$-Gaussian-Grothendieck problem and investigate the structure of the set
of all near optimizers along with stability estimates. In the latter case, we
show that this problem admits a Parisi-type variational representation and the
corresponding optimizer is weakly delocalized in the sense that its entries
vanish uniformly in a polynomial order.
