For the case $n=1$, it is well known that for the domain $\Omega = (a,b)$, the first eigenvalue for the $p$-Laplacian is given by
$$\lambda_1 = (p-1)(\frac{\pi_p}{b-a}),$$
where $$\pi_p = 2\int_0^1 \frac{ds}{\sqrt[p]{1-s^p}}.$$

For higher dimensions (n>1) where $p>1, p \neq 2$, the explicit expression is not known even for simple domains such as the sphere and the square. However, it is known that $\lambda_1$ is positive, simple, and the corresponding eigenfunction does not change sign.

Several numerical algorithms exist for computing the first eigenvalue of the $p$-Laplacian. For an iterative method, see http://arxiv.org/pdf/1011.3172.pdf. For an gradient descent algorithm based on the Rayleigh quotient formulation, see http://arxiv.org/pdf/1106.0602.pdf.