# A priori estimates for stable solutions to nonlinear elliptic equations

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In many elliptic variational problems, one usually needs the information on the stability or Morse index to get some good a priori estimates. In this talk I will review a common phenomenon about this estimate and some difficulties for semilinear elliptic equations. I will also discuss my joint work with J. Wei on stable solutions of Allen-Cahn equation, where we get a uniform second order estimate on clustering interfaces.