Qualitative Theory of Quasilinear Degenerate Elliptic Differential Equations

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We studied the subjects related to qualitative theory of nonlinear elliptic differential equations : (i) boundary value problems of quasilinear elliptic equations in a bounded domain ; (ii) blowup phenomena of seminilear parabolic equations ; (iii) solutions of ordinary differential equations derived from qualitative problem of elliptic equations in an unbounded domain. Our results are the following.1. Positive solutions of a class of nonlinear eigenvalue problems are investigated. For a quasilinear elliptic problem (^*)-div (φ(|∇u|) ∇u)=λf(x, u) in Ω, u=0 on ∂Ω, we assume asymptotic conditions on φ and f such as φ(t)〜t^<p_0-2>, f(x, t)〜t^<q_0-1> as t→0 andφ(t)〜t^<p_1-2>, f(x, t)〜t^<q_1-1> as t→∞. The combined effects of sub-nonlinearity (p_0>q_0) and super-nonlinearity (p_1<q_1) imply the existence of at least two positive solutions of (^*) for 0<λ<Λ.2. Concerning an initial value problem of semilinear parabolic equations with convex nonlinearities, we obtain sufficient conditions which determine the solution to blow up. The criteria are formulated in terms of super-solution and sub-solution of the related stational problem.3. A Sturm-Liouville equation on [α, ∞) is examined. (1) Higher order ordinary differential equations with general nonlinearities and determination of the Kiguraze class of positive sotuions. (2) Oscillation problem of first order 4-dimensional system of ordinary differential equations. (3) Eigenvalue problem of second order half-linear ordinary differential equations on [α, ∞).