Qualitative Theory of Nonlinear Elliptic Differential Equations

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We studied the subjects related to qualitative theory of nonlinear elliptic differential equations : (i) bound-ary value problems of quasilinear elliptic equations in a bounded domain ; (ii) oscillatory problem of ordi-nary differential equations which derived from qualitative problem of elliptic equations in an unbounded domain. Our results are the following.[1] The asymptotic behavior of eigenvalues and eigenfunctions of p-Laplace operator is investigated. We obtain the best constant of L*-Poincare inequality, and a limit equation which the limits of eigenvalues and eigenfuncitons satisfy in a weak sense.[2] A Sturm-Liouville equation on (a, *) is examined. Supposing a strongly nonoscillatory condition, we obtain a sequence of positive princial eigenvalues and the corresponding principal eigenfunctions.[3] Concering an initial value problem of parabolic equation, we obtain a new sufficient condition on the initial value which determines the solution to blow up. Furthermore, we investigate the asymtotic behavior of the solution at the blowing lip time.[4] We consider a second order half-linear differential equation on [a, *]. We take up the two classes of nonoscillatory solutions (i.e., princilal solutions and non principal solutions), and show that precise information can be drawn as to how the number of zeros of these solutions changes as lambda varies from zero to infinity.[5] A bifurcation problem or a nonlinear eigenvalue problem for degenerate quasilinear elliptic equations with Dirichelt boundary condition is studied. By virture of our estimates, we can apply the Leray-Shauder degree theory to our problem and obtain the bifurcation of nontrivial weak solutions.[6] Quasiperiodic solutions to Van der Pol type equations driven by two or more distinct freequency input signals are considered. The existence and uniqueness results are obtained from the viewpoint of numerical analysys.