Sign-changing Nonlinear Terms Research Articles

Positive radial solutions for elliptic equations with sign-changing nonlinear terms in an annulus

This paper deals with the existence and nonexistence of positive radial solutions of the elliptic equation where and , whose sign may change. We present new eigenvalue criteria for the existence and nonexistence to this problem. The discussion is based on the fixed point index theory in cones.

NONTRIVIAL SOLUTIONS OF NONLOCAL BOUNDARY VALUE PROBLEMS FOR NONLINEAR HIGHER-ORDER SINGULAR FRACTIONAL DIFFERENTIAL EQUATIONS

This paper deals with the existence and multiplicity of nontrivial solutions of nonlocal boundary value problems for nonlinear higher-order singular fractional differential equations with sign-changing nonlinear term. The main tool used in the proof is topological degree theory. Some examples explain that our results cannot be obtained by the method of cone theory.

Existence and Uniqueness of Solutions for a Class of Singular Nonlinear Two-Point Boundary Value Problems with Sign-Changing Nonlinear Terms

ABSTRACTIn this paper, a constructive proof under some conditions is presented for existence and uniqueness of the solutions to the singular problem with the boundary conditionsIn general, f(x,y(x)) is assumed to be nonsingular with respect to the independent variable x but it is allowed be singular with respect to y and moreover, f(x,y(x)) can be sign changing as well. The Picard iterative sequence is then constructed based on integral equation with the help of positive Green’s function. The convergence of this iterative sequence is controlled by an adjustable parameter so that it converges to the unique solution in the finite region in which is supposed to be nonnegative and bounded. Some examples also are given to demonstrate the trustworthiness of our constructive theory.

An Improved Fountain Theorem and Its Application

Abstract The main aim of the paper is to prove a fountain theorem without assuming the τ-upper semi-continuity condition on the variational functional. Using this improved fountain theorem, we may deal with more general strongly indefinite elliptic problems with various sign-changing nonlinear terms. As an application, we obtain infinitely many solutions for a semilinear Schrödinger equation with strongly indefinite structure and sign-changing nonlinearity.

Nontrivial Solutions of Higher-Order Nonlinear Singular Fractional Differential Equations with Fractional Multi-point Boundary Conditions

Abstract This paper deals with the existence and multiplicity of nontrivial solutions of fractional multi-point boundary value problems for higher-order nonlinear singular fractional differential equations with sign-changing nonlinear term. The main tool used in the proof is topological degree theory. Some examples are given to illustrate our main results.

Existence of Positive Solutions for Fourth-Order Boundary Value Problems with Sign-Changing Nonlinear Terms

the existence of positive solutions for a fourth-order boundary value problem with a sign-changing nonlinear term is investigated. By using Krasnoselskii’s fixed point theorem, sufficient conditions that guarantee the existence of at least one positive solution are obtained. An example is presented to illustrate the application of our main results.

Existence of positive solutions for fourth-order semipositone multi-point boundary value problems with a sign-changing nonlinear term

In this article, some new sufficient conditions are obtained by making use of fixed point index theory in cone and constructing some available integral operators together with approximating technique. They guarantee the existence of at least one positive solution for nonlinear fourth-order semipositone multi-point boundary value problems. The interesting point is that the nonlinear term f not only involve with the first-order and the second-order derivatives explicitly, but also may be allowed to change sign and may be singular at t = 0 and/or t = 1. Moreover, some stronger conditions that common nonlinear term f ≥ 0 will be modified. Finally, two examples are given to demonstrate the validity of our main results. 2000 Mathematics Subject Classification: 34B10; 34B18; 47N20.

Mountain pass theorem in ordered Banach spaces and its applications to semilinear elliptic equations

In this paper, we establish a mountain pass theorem in an ordered Banach space, which is a Riesz-Banach space such that the absolute value is continuous. Applying our theorem to semilinear elliptic equations with sign-changing nonlinear terms, we prove the existence of a positive solution.

Nontrivial solutions of singular fourth-order Sturm–Liouville boundary value problems with a sign-changing nonlinear term

This paper concerns the existence of nontrivial solutions for the following singular boundary value problem with a sign-changing nonlinear term: u ( 4 ) ( t ) = h ( t ) f ( t , u ( t ) , u ″ ( t ) ) , 0 < t < 1 , α 1 u ( 0 ) - β 1 u ′ ( 0 ) = δ 1 u ( 1 ) + γ 1 u ′ ( 1 ) = 0 , α 2 u ″ ( 0 ) - β 2 u ‴ ( 0 ) = δ 2 u ″ ( 1 ) + γ 2 u ‴ ( 1 ) = 0 , where h( t) is allowed to be singular at t = 0 and/or t = 1. Moreover, f ( t , x , y ) : [ 0 , 1 ] × R 2 → R is a sign-changing continuous function and may be unbounded from below with respect to x and y. By applying the topological degree of a completely continuous field and eigenvalue, some new results on the existence of nontrivial solutions for the above boundary value problem are obtained.

On eigenvalue intervals of higher-order boundary value problems with a sign-changing nonlinear term

In the paper, we shall consider the boundary value problem { u ( n ) + λ a ( t ) f ( t , u , u ′ , u ″ , u ( 3 ) , … , u ( n − 2 ) ) = 0 , n ≥ 2 , t ∈ ( 0 , 1 ) , u ( i ) ( 0 ) = 0 , 0 ≤ i ≤ n − 3 , α u ( n − 2 ) ( 0 ) − β u ( n − 1 ) ( 0 ) = 0 , γ u ( n − 2 ) ( 1 ) + δ u ( n − 1 ) ( 1 ) = 0 where λ > 0 , α , β , γ and δ are constants satisfying α , γ > 0 and β , δ ≥ 0 . Intervals of λ are determined to ensure the existence of a positive solution of the boundary value problem according to the signs of a ( t ) and f .

Nontrivial solutions for higher-order m-point boundary value problem with a sign-changing nonlinear term

This paper studies the existence of nontrivial solutions for a class of higher-order m-point singular boundary value problems based on the topological degree of a completely continuous field, the first eigenvalue and its corresponding eigenfunction of a special linear operator. The nonlinear term in the boundary value problems is sign-changing and may be unbounded from below.

POSITIVE SOLUTIONS OF BOUNDARY VALUE PROBLEMS FOR p-LAPLACIAN WITH SIGN-CHANGING NONLINEAR TERMS

By using the fixed point index theory, we investigate the existence of at least two positive solutions for p-Laplace equation with sign-changing nonlinear terms ('p(u 0 )) 0 +a(t)f(t,u(t),u 0 (t)) = 0, subject to some boundary conditions. As an application, we also give an example to illustrate our results.

Positive Solution to a Singular -Laplacian BVP with Sign-Changing Nonlinearity Involving Derivative on Time Scales

Let be a time scale such that . By using a monotone iterative method, we present some existence criteria for positive solution of a multiple point general Dirichlet-Robin BVP on time scales with the singular sign-changing nonlinearity. These results are even new for the corresponding differential and difference equation as well as in general time scales setting. As an application, an example is given to illustrate the results. The interesting point here is that the sign-changing nonlinear term is involved with the first-order derivative explicitly, and the singularity may occur at , , and .

Nontrivial solutions of m-point boundary value problems for singular second-order differential equations with a sign-changing nonlinear term

This paper concerns the existence of nontrivial solutions for the following singular m -point boundary value problem with a sign-changing nonlinear term { ( L u ) ( t ) + h ( t ) f ( t , u ) = 0 , 0 < t < 1 , u ( 0 ) = 0 , u ( 1 ) = ∑ i = 1 m − 2 a i u ( ξ i ) , where ( L u ) ( t ) = ( p ̃ ( t ) u ′ ( t ) ) ′ + q ( t ) u ( t ) , 0 < ξ 1 < ξ 2 < ⋯ < ξ m − 2 < 1 , a i ∈ [ 0 , + ∞ ) , h ( t ) is allowed to be singular at t = 0 , 1 , and f : [ 0 , 1 ] × ( − ∞ , + ∞ ) → ( − ∞ , + ∞ ) is a sign-changing continuous function and may be unbounded from below. By applying the topological degree of a completely continuous field and the first eigenvalue and its corresponding eigenfunction of a special linear operator, some new results on the existence of nontrivial solutions for the above singular m -point boundary value problem are obtained. An example is then given to demonstrate the application of the main results. The work improves and generalizes the main results of [G. Han, Y. Wu, Nontrivial solutions of singular two-point boundary value problems with sign-changing nonlinear terms, J. Math. Anal. Appl. 325 (2007) 1327–1338; J. Sun, G. Zhang, Nontrivial solutions of singular superlinear Sturm-Liouville problem, J. Math. Anal. Appl. 313 (2006) 518–536].

Nontrivial solutions of singular two-point boundary value problems with sign-changing nonlinear terms

The singular two-point boundary value problem − u ″ ( t ) = h ( t ) f ( u ( t ) ) , t ∈ ( 0 , 1 ) ; u ( 0 ) = u ( 1 ) = 0 is considered under some conditions concerning the first eigenvalue corresponding to the relevant linear problem, where h is allowed to be singular at both t = 0 and t = 1 . Moreover, f : ( − ∞ , + ∞ ) → ( − ∞ , + ∞ ) is a sign-changing function and not necessarily bounded from below. By computing the topological degree of an completely continuous field, the existence results of nontrivial solutions are established.

Positive solutions of second-order boundary value problems with sign-changing nonlinear terms

In this paper the existence results of positive solutions are obtained for second-order boundary value problem −u″=f(t,u),t∈(0,1), u(0)=u(1)=0, where f∈C([0,1]× R +) , whose sign may change. The proof is based on the fixed point index theory in cones.

COMPUTATIONS OF THE COHOMOLOGY GROUPS WITH APPLICATIONS TO ASYMPTOTICALLY LINEAR BEAM EQUATIONS AND NONCOOPERATIVE ELLIPTIC SYSTEMS1*

ABSTRACT We develop a method to compute the -cohomology critical groups for strongly indefinite functionals. The abstract theory is applied to prove the existence of (multiple) nontrivial solutions for asymptotically linear beam equations and noncooperative elliptic systems with unbounded sign-changing nonlinear terms and with resonance both at infinity and at zero. This work was partially supported by NSFC (10001019) and the Swedish Institute, the Swedish Royal Academy of Sciences of Sweden.