Small particles or islands bonded to a substrate can be profoundly influenced by both interfacial and elastic driving forces that tend to have opposing influences on the apparent wetting behavior. The superposition of these two driving forces can therefore lead to a rich set of particle properties, most notably their equilibrium shapes. Here we present a variational analysis leading directly to an Euler-Lagrange equation that can be solved to yield the equilibrium shapes of partially wetting particles as a function of their size, interface energy densities, and elastic interaction with a rigid substrate. The solutions are used to gain insight into the variables that most significantly influence the equilibrium morphology, and to derive the approximate driving force for surface area reduction by coarsening among a dispersion of unequally sized particles. The relatively simple analytical model can also form a foundation upon which more realistic numerical simulations may be built and compared.