Structure of solution spaces for singular partial differential equations.

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(1) We prove the existence of solutions for semilinear elliptic equations with the singular weight on the boundary. We prove the existence of a smooth positive solution and infinitely many sign-changing solutions.(2) We prove that the bifurcation occurs for one-dimensional p-Laplace equations with singular weights on the boundary.(3) For degenerate parabolic equations with the infinity Laplacian, we prove that solutions decay as time tends to infinity and investigate the decay order in detail.