Fix $p>1$, not necessarily integer, with $p(d-2)<d$. We study the $p$-fold self-intersection local time of a simple random walk on the lattice $Z^d$ up to time $t$. This is the $p$-norm of the vector of the walkers local times, $ell_t$. We derive precise logarithmic asymptotics of the expectation of $exp{theta_t |ell_t|_p}$ for scales $theta_t>0$ that are bounded from above, possibly tending to zero. The speed is identified in terms of mixed powers of $t$ and $theta_t$, and the precise rate is characterized in terms of a variational formula, which is in close connection to the {it Gagliardo-Nirenberg inequality}. As a corollary, we obtain a large-deviation principle for $|ell_t|_p/(t r_t)$ for deviation functions $r_t$ satisfying $t r_tggE[|ell_t|_p]$. Informally, it turns out that the random walk homogeneously squeezes in a $t$-dependent box with diameter of order $ll t^{1/d}$ to produce the required amount of self-intersections. Our main tool is an upper bound for the joint density of the local times of the walk.
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