Studies on blow-up analysis of critical variational problems and qualitative properties of solutions caused by blow-up

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In this research, we studied various nonlinear elliptic equations associated with variational problem of so called "critical type". By using the intrinsic scale invariance of the equations involved, we made "blow up analysis" of non-compact solution sequences to the equations and by exploiting the explicit structure of solution set of the limit equations, we obtain various qualitative properties of blowing up solutions, such as asymptotic nondegeneracy, asymptotic uniqueness, and so on. Also we studied spectral property of blowing up solutions and obtain several results on this matter.