  A wide variety of articles, starting with the famous paper (Gidas, Ni and
Nirenberg in Commun. Math. Phys. 68, 209-243 (1979)) is devoted to the
uniqueness question for the semilinear elliptic boundary value problem
-{\Delta}u={\lambda}u+u^p in {\Omega}, u>0 in {\Omega}, u=0 on the boundary of
{\Omega}, where {\lambda} ranges between 0 and the first Dirichlet Laplacian
eigenvalue. So far, this question was settled in the case of {\Omega} being a
ball and, for more general domains, in the case {\lambda}=0. In (McKenna et al.
in J. Differ. Equ. 247, 2140-2162 (2009)), we proposed a computer-assisted
approach to this uniqueness question, which indeed provided a proof in the case
{\Omega}=(0,1)x(0,1), and p=2. Due to the high numerical complexity, we were
not able in (McKenna et al. in J. Differ. Equ. 247, 2140-2162 (2009)) to treat
higher values of p. Here, by a significant reduction of the complexity, we will
prove uniqueness for the case p=3.
