Are there compounds that melt under their own weight?
Sure. Even discounting non-equilibrium conditions, any substance which expands upon freezing will by symmetry melt under pressure, even if the pressure is caused by supporting its own weight. There's a simple example: water! A sufficiently tall self-supporting block of ice standing on a flat surface will create enough pressure at its bottom that it could melt in temperatures between approximately $-21.9\ ^\circ \mathrm{C}$ and $0\ ^\circ \mathrm{C}$, according to the phase diagram for water. This has very important implications for the study of glaciers, for example. However, this process requires a very large amount of pressure, and thus a very tall block of ice. Using an approximate value for the freezing point depression under pressure of $0.0072\ ^\circ \mathrm{C\ atm^{-1}}$, it would take an ice block about $7.5\ \mathrm{km}$ tall for the base to melt at a temperature of $-5\ ^\circ \mathrm{C}$ or higher.

I thought indium was among the substances which expanded upon freezing, but it actually turns out to contract by 2.5% when frozen, so this mechanism would not work (though there could be something else to consider).  Some other materials which could theoretically melt under their own weight in some regime of temperatures and pressures are gallium, bismuth and silicon. There might be some very specific conditions under which this effect could be achieved in a column of around $20\ \mathrm{cm}$, though pressure usually only slightly alters the freezing point, so it's likely that the material would have to be kept only slightly under its melting point at standard pressure for this to happen.