Studies on the Structure of Solutions of Degenerate Quasilinear Elliptic Equations

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We have considered quasilinear degenerate elliptic equations which are equal to p-Laplacians asymptotically at tile origin and infinity. Some results concerning to the aspects of solutions for the equations of this type are obtained as are listed below.1. First we have obtained the structure of the branches of positive solutions bifurcated from the trivial solution and tile infinity solution. Namely, positive solutions bifurcate from the zero solution and the infinity solution at the first eigenvalues of the p-Laplacians which the considered equation are equal to aymptotically at zero and infinity respectively. Furthermore, in the case when the growth orders of the principal part of the equation at zero and infinity, these branches are identical.2. Next the asymptotic behavior of eigenvalues and eigenfunctions of p-Lapalace operators as the power p tends to infinity has been investigated. We have obtained the best constant of Poincare's inequality at p=*, and further, given a limit eigenvalue problem which characterize tile limits of these eigenvalues and eigenfunctions. Althogh the equation ill the limit eigenvalue problem is a fully nonlinear elliptic equation, a pair of the limits of eigenvalues and eigenfunctions solves this limit eigenvalue problem in the sense of viscosity.3. Finally we have considered the equation defined in the whole space instead of a bounded domain. In this case, tile similar results as are stated in I.are also valid under some assumptions on the potential part of the nonlinear operator. It is our problem to loose these restrictions hereafter.