Minimal strictly stable solutions of -Δu=f(u) in smooth uniformly convex planar domains can have nonconvex superlevel sets for f(u)=e^u or f(u)=(a+u)^p, answering Brezis's open question negatively.
Brezis,Is there failure of the inverse function theorem?, Morse theory, minimax theory and their applications to nonlinear differential equations, 23–33, New Stud
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Strictly stable solutions in uniformly convex planar domains may have nonconvex superlevel sets
Minimal strictly stable solutions of -Δu=f(u) in smooth uniformly convex planar domains can have nonconvex superlevel sets for f(u)=e^u or f(u)=(a+u)^p, answering Brezis's open question negatively.