Number Of Positive Solutions Research Articles

Positive solutions of even order system periodic boundary value problems

We study a class of even order system boundary value problems with periodic boundary conditions. A series of criteria are obtained for the existence of one, two, any arbitrary number, and even a countably infinite number of positive solutions. Criteria for the nonexistence of positive solutions are also derived. As for the second order case, our results extend, improve, and supplement those in the literature for scalar and system boundary value problems. Several examples are given to demonstrate the applications. Moreover, we obtain conditions for system periodic boundary value problems of a different form to have nontrivial solutions by transforming our main results to such problems.

On S-shaped bifurcation curves for a class of perturbed semilinear equations

By making use of bifurcation analysis and continuation method, the authors discuss the exact number of positive solutions for a class of perturbed equations. The nonlinearities concerned are the so-called convex-concave functions and their behaviors may be asymptotic sublinear or asymptotic linear. Moreover, precise global bifurcation diagrams are obtained.

On the evolution and qualitative behaviors of bifurcation curves for a boundary value problem. II

We study the evolution and qualitative behaviors of bifurcation curves of positive solutions for { − u ″ ( x ) = λ f q , p ( u ) = λ ( u q ( 1 − sin u ) + u p ) , − 1 < x < 1 , u ( − 1 ) = u ( 1 ) = 0 , where λ > 0 is a bifurcation parameter, q < 1 is a positive bifurcation parameter, and p ≥ 1 is an evolution parameter. We prove that, for given q < 1 , there exist numbers p ∗ ( q ) > p ∗ ( q ) > 1 such that, on the ( λ , ‖ u ‖ ∞ ) -plane, the bifurcation curve has exactly one turning point where the curve turns to the left for p > p ∗ ( q ) , it has at least three turning points for 1 < p < p ∗ ( q ) , and it has infinitely many turning points for p = 1 . Hence we are able to determine the (exact) number of positive solutions. In particular we give complete descriptions of the structure of bifurcation curves when p > p ∗ ( q ) . Our results extend some results of Wang [S.-H. Wang, On the evolution and qualitative behaviors of bifurcation curves for a boundary value problem, Nonlinear Anal. 67 (2007) 1316–1328] from q = 1 to 0 < q ≤ 1 .

On the evolution and qualitative behaviors of bifurcation curves for a boundary value problem

We study the evolution and qualitative behaviors of bifurcation curves of positive solutions for { − u ″ ( x ) = λ ( u ( 1 − sin u ) + u p ) , − 1 < x < 1 , u ( − 1 ) = u ( 1 ) = 0 , where λ > 0 is a bifurcation parameter and p ≥ 1 is an evolution parameter. On the ( λ , ‖ u ‖ ∞ ) -plane, we prove that the bifurcation curve has exactly one turning point where the curve turns to the left for p > 2 , it is a monotone curve for p = 2 , it has at least two turning points for 1 < p ≤ p ̃ ≡ 1 + 2 cos 1 − sin 1 1 − 2 cos 1 + sin 1 ≈ 1.629 , and it has infinitely many turning points for p = 1 . Hence we are able to determine the (exact) number of positive solutions. In particular we give complete descriptions of the structure of bifurcation curves when p ≥ 2 . We also give some numerical simulations of bifurcation curves for p ≥ 1 .

Nonlinear elliptic problems on singularly perturbed domains

We consider how the shape of a domain affects the number of positive solutions of a nonlinear elliptic problem. In fact, we show that if a bounded domain Ω is sufficiently close to a union of disjoint bounded domains Ω1,…, Ωm, the number of positive solutions of a nonlinear elliptic problem on Ω is at least 2m −1.

Exact multiplicity of positive solutions in semipositone problems with concave-convex type nonlinearities

We study the existence, multiplicity, and stability of positive solutions to: u 00 (x) = f (u(x)) for x 2 ( 1; 1); > 0; u( 1) = 0 = u(1); where f : (0;1) ! R is semipositone (f(0) 0, we obtain the exact number of positive solutions as a function of f(t)=t and establish how the positive solution curves to the above problem change. Also, we give examples where our results apply. This work extends the work in (1) by giving a complete classication of positive solutions for concave-convex type nonlinearities. (t) f(t) t2 and (tf 0 (t) f(t)) 0 = tf 00 (t) with f(0) 0 for t 2 (0; t1) ( (t2;1) and (f(t)=t) 0 < 0 for t 2 (t1; t2)

Multiplicity of positive solutions to semilinear elliptic boundary value problems

We study semilinear elliptic boundary value problems of one parameter dependence where the number of positive solutions is discussed. Our main purpose is to characterize the critical value given by the infimum of such parameters for which positive solutions exist. Our approach is based on super‐ and sub‐solutions, and relies on the topological degree theory on the positive cones of ordered Banach spaces. A concrete example is also presented.

A note on the number of positive solutions of some nonlinear elliptic problems

A note on the number of positive solutions of some nonlinear elliptic problems

About the number of positive solutions of Neumann problems

About the number of positive solutions of Neumann problems

The set of positive solutions of semilinear equations in large balls

SynopsisThe exact number of positive solutions of Δu + f(u) = 0 on finite balls in ℝN is determined. The assumptions about f(u) are similar to those imposed by Serrin and the second author in a previous study of uniqueness of the positive solution when the spatial domain is all of ℝN (see [7, 8]). For finite balls of sufficiently large radius it is shown here that there are exactly two positive and, hence, radial solutions. To this end, we first prove the linear nondegeneracy of the positive solution of ℝN. This is obtained by applying the technique of monotone separation of graphs [7] to the linearised equations. Somewhat sharper estimates are required here (see Part I, Section 2).

The Role of Seating Types in Egress Difficulty Experienced by Older Adults: A Biomechanical Analysis

Videotape analysis of biomechanical movements involved in chair egress was used to assess the efficacy of the Warren Chair, a non-electromotive assistive chair. Persons experiencing chair egress difficulties were videotaped egressing both the Warren Chair and their own standard chair. Compared to standard chairs, Warren Chair egresses were easier, required less time, were more ergonomically efficient and provided more postural balance at the completion of the egress. The authors conclude that the Warren Chair provides a number of positive solutions for the problems experienced by persons in chair egress.

Asymptotically linear, convex boundary value problems

In This paper results of Amann [1] and Ambrosetti [4] concerning with the number of positive solutions of an aysmptotically linear, convex boundary value problem are improved.

Uniqueness for sublinear boundary value problems

This paper presents uniqueness theorems for positive solutions of the semilinear second order elliptic boundary value problem (1) below with a nonnegative nonlinearity F(x, zu) which is monotonic and sublinear (Eq. (2)) in w. The proofs of the theorems use the methods developed by Krasnosel’skii [l, Sections 6.1 and 7.2.61 and Urysohn [2] f or nonlinear operators together with the maximum principle of Hopf [3] f or elliptic difkrential operators, Since use is made of the maximum principle, we consider only classical (positive) solutions of the boundary value probiem, and the only smoothness assumptions necessary for the given functions appearing in the probIem are that the coefhcients of the differential operator be bounded in the domain J2 in which the solution is sought. In order to apply the techniques of Urysohn and Krasnosel’skii, however, it is necessary to make a careful study of the boundary of !ZJ and the behavior of the solutions near the boundary (cf. Theorem 2). The uniqueness theorem states that under certain smoothness conditions (see especially Corollary 2.2), (1) h as either at most 01x positive solution or an infinite number of positive solutions, each of which is a scalar multiple of any other. For domains whose boundaries satisfy the conditions imposed below, our results include the uniqueness results of Keller [4, 5, 61, Kdler and Cohen [7], Cohen [S], and Shampine [9], for self-adjoint elliptic operators. In contmst to the results just cited, our method of proof does not require one to assume the existence of solutions (eigenvalues and eigenfunctions) for associated linear problems or the differentiability of the nonlinearityfi: The technique used here has been applied to a closely related problem in [lo]. For ordinary differential equations, the uniqueness of the positive solutions for sublinear ~nonlinearities is proven by Pimbley [l l] (see also [12] and [l, Section 7.4.41). The existence