Semilinear elliptic equations on all of R^N typically have solutions that decay at infinity but never vanish — the tails stretch to infinity. But when the nonlinearity is sublinear and changes sign sharply — positive in a bounded domain, negative outside — something different happens.
All solutions have compact support (arXiv:2603.11766). Every solution, whether ground state or nodal (sign-changing), vanishes identically outside a bounded region. The support of the ground state is starlike and has Lipschitz boundary. As the exponent approaches 2 (the linear threshold), the support geometry converges to a well-defined limit connected to Serrin-type torsion problems.
The structural observation: the equation is posed on all of R^N, an unbounded domain, but every solution spontaneously localizes. The compact support is not imposed — it emerges from the interplay between sublinearity and sign change. The sign change creates a region where the nonlinearity is repulsive (negative), and the sublinearity means the solution lacks the algebraic strength to penetrate this repulsion. The solution doesn't decay; it stops. The free boundary between solution and void is a geometric object with its own regularity theory.