On the structure of singular solutions for degenerate elliptic equations

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1.On the structue of singular solutions for degenerate ellptic equations :(1)We studied p-harmomic equation with strong positive nonlinear terms in the right-hand side systematically, and established the unique existence results of minimal solutions. Moreover we studied very well linearized operators at the minimal solutions. Proving fundamental properties of the linearized operators, we made clear the coercivity and positivity of the operator and applied them to construct the theory of Bifurcation.(2)We studied various type of generalized Hardy-Sobolev-Rellich inequalities, and improved them by finding out sharp missing terms. We applied them to the study of Blow-up solutions.2.On the regularity of solutions for genuinely degenerated elliptic equations :The existence of bounded solutions for degenerate elliptic equations were studied. To study further regularities, multiplicative Sobolev inequalities with weghts were established.3.On the variational problems with critical nonlinear terms and singular solutions :Using the results in the above, singular variational problems were investigated.4.The potential theory for degenerate elliptic operators :Multi-parabolic operator was studied and generalized mean-value property was established.