New entire solutions to semilinear elliptic equations

We will survey some recent results on the construction of entire solutions of semilinear elliptic equations. We will mostly focus on the construction of families of solutions to the Allen–Cahn equation of phase transitions, whose level sets suitable scaled concentrate around a given minimal surface. To do so, we introduce an infinitedimensional form of Lyapunov–Schmidt reduction suitable for this and several related questions.