Hiroyuki Usami

• Gifu University

In this paper the half-linear differential equation with one-dimensional p-Laplacian (p=α+1),(|u′|αsgnu′)′=α(λα+1+b(t))|u|αsgnu,t≥t0, is considered, where α and λ are positive constants. It is proved that if the function b(t) is absolutely integrable on [t0,∞), then the above equation has two nonoscillatory solutions u+(t) and u−(t) such that u±(t)...

In this paper we discuss the existence and asymptotic behavior of strongly increasing solutions of quasilinear ordinary differential equations of the form D ( α n , α n − 1 , … , α 1 ) x = p ( t ) | x | β sgn x , t ≥ a . $$\begin{array}{} \displaystyle D(\alpha_n, \alpha_{n-1}, \dots, \alpha_1)x = p(t)|x|^{\beta}\text{ sgn } x, \quad t \geq a. \end...

In this paper we discuss the existence and asymptotic behavior of Kneser solutions of nth-order two term quasilinear ordinary differential equations which are generalizations of the Emden-Fowler equation. It will be shown that there is a simple and significant difference between the super-homogeneous case and the sub-homogeneous case for the existe...

This paper studies an inverse problem to determine a nonlinearity of an autonomous equation from blow-up time of solutions to the equation. Firstly we prove a global continuation result showing that a nonlinearity realizing blow-up time for large initial data can be continued in the direction of smaller data as long as the blow-up time is Lipschitz...

We determine asymptotic forms of slowly decaying positive solutions of quasilinear ordinary differential equations related to so-called p-Laplace equation.

We study an inverse problem to determine a nonlinearity of an autonomous equation from a blow-up time of solutions of the equation. A local well-posedness of the inverse problem near a nonlinearity of the type u1+σ, σ > 0, is established. The paper also suggests that the inverse problem has a good, mathematical structure from a viewpoint of the Wie...

Let $\Omega= \{x\in\bbfR^N: r_0\le|x|< 1\}$ with $N\ge 2$ and $r_0\in (0,1)$. We study a kind of geometric oscillatory and asymptotic behaviour near $|x|= 1$ of all radially symmetric solutions $u=u(x)$ of the $p$-Laplace partial differential equation $$(P): -\text{div}(|\nabla u|^{p-2}\nabla u)= f(|x|)|u|^{p-2} u\quad\text{in }\Omega,$$ $u=0$ on $...

We consider fourth order quasilinear ordinary differential equations. Firstly, we classify positive solutions into four types according to their asymptotic properties. Then we derive existence theorems of positive solutions belonging to each type. Using these results, we can obtain an oscillation criterion, which is our main objective. Moreover, ap...

We consider quasilinear ordinary differential equations with sub-homogeneity near infinity. A necessary and sufficient condition is obtained for the equations to have slowly decaying positive solutions. Asymptotic forms of such positive solutions are established. As an application of these results, we obtain Liouville-type theorems for quasilinear...

We show how one-dimensional generalized Riccati-type inequalities can be employed to analyse asymptotic behaviour of solutions of elliptic problems. We give Liouville-type theorems as well as necessary conditions for the existence of solutions of specified asymptotic behaviour of nonlinear elliptic problems.

In this paper we consider positive unbounded solutions of second order quasilinear ordinary differential equations. Our objective is to determine the asymptotic forms of unbounded solutions. An application to exterior Dirichlet problems is also given.

We consider singular boundary value problems to quasilinear ordinary differential equations with singular nonlinearities. We give sufficient conditions for such problems to have a positive solution. Moreover, for a typical case we can show also the uniqueness of the solutions.

In this paper we consider positive solutions of second order quasilinear ordinary differential equations with singular nonlinearities. We obtain asymptotic equivalence theorems for asymptotically superlinear solutions and decaying solutions. By using these theorems, exact asymptotic forms of such solutions are determined. Furthermore, we can establ...

Asymptotic forms are determined for positive solutions which are called weakly increasing solutions to quasilinear ordinary differential equations.

We establish oscillation criteria for solutions of quasilinear second order elliptic equations. We do not impose any additional conditions on the nonlinear terms except for the continuity. In particular, we can characterize the oscillation property of every solution for autonomous equations.

Existence theory of nonoscillatory solutions to binary elliptic systems is developed. The results are based on comparison principles and asymptotic theory for one-dimensional problems. Indiana University Mathematics Journal

Oscillation criteria for fourth-order quasilinear ordinary differential equations are obtained. An application to binary semilinear elliptic systems is also given.

We obtain asymptotic forms of positive solutions of third-order Emden–Fowler equations. These results improve earlier results for asymptotic behavior of positive solutions considerably.

This paper treats second-order, semilinear systems of the form Δu=p(x)v α ,Δv=q(x)u β ,x∈ℝ N where α,β>0 are constants satisfying αβ>1, and p,q∈C(ℝ N ;(0,∞)). We obtain a Liouville type theorem for nonnegative entire solutions of this system.

We study asymptotic behavior of solutions of a class of second order quasilinear ordinary differential equations. All solutions are classified into six types by means of their asymptotic behavior. Necessary and/or sufficient conditions are given for such equations to possess a solution of each of the six types.

We give asymptotic forms of positive solutions of second-order quasilinear ordinary differential equations. We obtain further the uniqueness of positive decaying solutions.

Oscillation criteria are obtained for quasilinear elliptic equations of the form (E)below. We are mainly interested in the case where the coefficient function oscillates near infinity. Generalized Riccati inequalities are employed to establish our results.

This paper treats the quasilinear elliptic inequality where N ≥ 2, m > 1, σ > m − 1, and p :ℝ N → (0, ∞ ) is continuous. Sufficient conditions are given for this inequality to have no positive entire solutions. When p has radial symmetry, the existence of positive entire solutions can be characterized by our results and some known results.

The semilinear elliptic equation (FORMULA PRESENTED) is considered, where (FORMULA PRESENTED) Our main objective is to present sufficient conditions and necessary and sufficient conditions for equation (1.1) to have positive radial entire solutions u(|x|) satisfying the asymptotic condition that (FORMULA PRESENTED) or the stronger asymptotic condit...

Our main objective is to prove the existence of infinitely many pairs (u1, u2) of positive solutions of quasilinear elliptic differential equations throughout exterior domains Ωα ∈ RN, N ≥ 2, of the type where x = (x1, …, XN), Δu = (∂u/∂x1,…,∂u/∂xN), and Δ = ∇ · ∇. Each pair has the property that u1(x)/u2(x) has uniform limit zero in Ωα as |x| → ∞....

On considere l'equation elliptique a 2 dimensions Δu=f(x,u,⊇u), x∈R 2 , ou x=(x 1 ,x 2 ) et f(x,u,p) est une fonction continue non negative sur R 2 ×R + XR 2 . On considere l'existence de solutions positives definies dans tout R 2

This paper treats the quasilinear elliptic inequality div( Du m 2Du) p(x)u x N where N 2, m 1, m 1, and p: N (0 ) is continuous. Sufficient conditions are given for this inequality to have no positive entire solutions. When p has radial symmetry, the existence of positive entire solutions can be characterized by our results and some known results.

We give some upper bounds for positive solutions of nonlinear elliptic inequalities specified below. Comparison principles are considerably used to analyze these problems.