Sign-changing Nonlinearity Research Articles

Locating peaks of a Schrödinger equation with sign-changing nonlinearity

In this paper, we study the concentration phenomenon of a positive ground state solution of a nonlinear Schrödinger equation on R N . The coefficient of the nonlinearity of the equation changes sign. We prove that the solution has a maximum point at x 0 ∈ Ω + = { x ∈ R N : Q ( x ) > 0 } where the energy attains its minimum.

POSITIVE SOLUTIONS TOp(x)-LAPLACIAN–DIRICHLET PROBLEMS WITH SIGN-CHANGING NON-LINEARITIES

AbstractConsider thep(x)-Laplacian–Dirichlet problem with sign-changing non-linearity of the formwhere Ω ⊂ ℝNis a bounded domain,p∈C0(Ω) and infx∈Ωp(x) > 1,m∈L∞(Ω) is non-negative,f: ℝ → ℝ is continuous andf(0) > 0, the coefficienta∈L∞(Ω) is sign-changing in (Ω). We give some sufficient conditions to assure the existence of a positive solution to the problem for sufficiently small λ > 0. Our results extend the corresponding results established in thep-Laplacian case to thep(x)-Laplacian case.

Periodic solutions for functional differential equations with sign-changing nonlinearities

We establish some eigenvalue criteria for the existence of non-trivial T-periodic solutions of a class of first-order functional differential equations with a nonlinearity f(x). The nonlinear term f(x) can take negative values and may be unbounded from below. Conditions are determined by the relationship between the behaviour of the quotient f(x)/x for x near 0 and ±∞ and the smallest positive characteristic value of an associated linear integral operator. This linear operator plays a key role in the proofs of the results and its construction is non-trivial. Applications to related eigenvalue problems are also discussed. The analysis mainly relies on the topological degree theory.

Symmetric positive solutions of fourth order integral BVP for an increasing homeomorphism and homomorphism with sign-changing nonlinearity on time scales

In this paper, we consider the eigenvalue problems for fourth order integral boundary value problems on time scales for an increasing homeomorphism and homomorphism with sign changing nonlinearities. By using a fixed point index theorem, we give the existence of eigenvalue intervals in which there exist one symmetric positive solution to the problem. An example is also given to demonstrate the main results.

Nonexistence and existence of positive solutions for second order singular three-point boundary value problems with derivative dependent and sign-changing nonlinearities

This paper discusses the nonexistence and the existence of positive solutions for second order singular three-point boundary value problems when the nonlinear term f(t,x,y) is sign-changing and may be singular at t=0, x=0, y=0.

The existence of multiple positive solutions for singular functional differential equations with sign-changing nonlinearity

In this paper, we study the existence of multiple positive solutions for boundary value problems based on second-order functional differential equations with the form { y ″ ( t ) + f ( t , y ( t − τ ) ) = 0 , ∀ t ∈ ( 0 , 1 ) ∖ { τ } , y ( t ) = η ( t ) , ∀ t ∈ [ − τ , 0 ] , y ( 1 ) = 0 where 0 < τ < 1 and f : ( 0 , 1 ) × ( 0 , + ∞ ) → ( − ∞ , + ∞ ) is continuous, may be singular at t = 0 , 1 , y = 0 and takes negative values. By applying the fixed point index theorem, we obtain the conditions for the existence of at least two and of three positive solutions. An example to illustrate our results is given.

On positive solutions for $p$-Laplacian systems with sign-changing nonlinearities

We consider the existence and multiplicity of positive solutions to the quasilinear system $$ \begin{cases} -\Delta_{p_{i}}u_{i} = \mu_{i}a_{i}(x)f_{i}(u_{1},\dots,u_{n})~\text{in}~\Omega,\;i=1,\dots,n, \\[1pt] u_{i} = 0~\text{on}~\partial \Omega , \end{cases} $$ where $\Omega$ is a bounded domain in $\mathbb{R}^{N}$ with a smooth boundary $\partial \Omega$, $\Delta_{p_{i}}u_{i}={\rm div}(|\nabla u_{i}|^{p_{i}-2}\nabla u_{i})$, $p_{i}>1$, $\mu_{i}$ are positive parameters, and $f_{i}$ are allowed to change sign.

Positive pseudo-symmetric solutions for higher order differential equation boundary value problems with sign changing nonlinearities

In this paper, a new fixed point theorem in cone is applied to obtain the existence of two positive pseudo-symmetric solution for the 2 n-order differential equation boundary value problem y ( 2 n ) ( t ) = f t , y , y ″ , … , y 2 ( n - 1 ) , 0 ⩽ t ⩽ 1 , y ( 2 i ) ( 0 ) = 0 , y ( 2 i ) ( 1 ) = y ( 2 i ) ( η ) , 0 ⩽ i ⩽ n - 1 , where f is allowed to change sign, η ∈ ( 0 , 1 ) .

Higher order multi-point boundary value problems with sign-changing nonlinearities and nonhomogeneous boundary conditions

We study classes of nth order boundary value problems consisting of an equation having a sign-changing nonlinearity f(t, x) together with several different sets of nonhomogeneous multi-point boundary conditions. Criteria are established for the existence of nontrivial solutions, positive solutions, and negative solutions of the problems under consideration. Conditions are determined by the behavior of f(t, x)/x near 0 and ±∞ when compared to the smallest positive characteristic values of some associated linear integral operators. This work improves and extends a number of recent results in the literature on this topic. The results are illustrated with examples.

Multiple solutions for fourth order m-point boundary value problems with sign-changing nonlinearity

Using a fixed point theorem in ordered Banach spaces with lattice structure founded by Liu and Sun, this paper investigates the multiplicity of nontrivial solutions for fourth order m-point boundary value problems with sign-changing nonlinearity. Our results are new and improve on those in the literature.

Positive solutions of sublinear Sturm–Liouville problems with changing sign nonlinearity

In this paper, the following nonlinear Sturm–Liouville problem {−(p(x)u′(x))′+q(x)u(x)=λf(x,u(x)),0≤x≤1,α0u(0)+β0u′(0)=0,α1u(1)+β1u′(1)=0 is discussed by topological methods. In the case that the nonlinear term is non-singular or singular, a global structure of the positive solution set of the above problem is obtained, and the existence of positive solutions is proved under the condition that the nonlinear term is allowed to change sign.

Multiplicity of positive solutions for a nonlinear m-point BVP with sign changing nonlinearity on time scales

In this paper, by using fixed point theorems in cones, the existence of positive solutions is considered for a nonlinear m-point boundary value problem for the following second-order dynamic equation on time scales $$\begin{array}{l}(\phi(u^{\Delta}))^{\nabla}+a(t)f(t,u(t))=0,\quad t\in(0,T),\\\noalign{\vspace{2mm}}\displaystyle u(0)=\sum_{i=1}^{m-2}a_{i}u(\xi_{i}),\qquad \phi(u^{\Delta}(T))=\sum_{i=1}^{m-2}b_{i}\phi(u^{\Delta}(\xi_{i})),\end{array}$$ where φ:R⟶R is an increasing homeomorphism and positive homomorphism with φ(0)=0. The nonlinear term f may change sign. We obtain several existence theorems of positive solutions for the above boundary value problems. We should point out that the above equation we studied is same as that in Han and Jin (Communications in Nonlinear Science and Numerical Simulation, 2009), but the methods is different from Han and Jin (Communications in Nonlinear Science and Numerical Simulation, 2009), we generalize and improve the results (Han and Jin, Communications in Nonlinear Science and Numerical Simulation, 2009). As an application, a typical example to demonstrate our results is given.

Positive solutions of nonlinear m-point BVP for an increasing homeomorphism and positive homomorphism with sign changing nonlinearity on time scales

In this paper, by using fixed point theorems in cones, some new existence criteria for positive solutions of the nonlinear m-point boundary value problem with an increasing homeomorphism and positive homomorphism are presented. The nonlinear term f is allowed to change sign. In particular, our criteria generalize and improve some known results [20] and the obtained conditions are different from related literature [16,17,23]. As an application, one representative example to demonstrate our results are given.

Positive solutions for one-dimensional [formula omitted]-Laplacian boundary value problems with sign changing nonlinearity

This paper deals with the existence of positive solutions for the one-dimensional p -Laplacian ( ϕ p ( u ′ ) ) ′ + f ( t , u , u ′ ) = 0 , t ∈ [ 0 , 1 ] , subject to the boundary value conditions: u ′ ( 0 ) = ∑ i = 1 n α i u ′ ( ξ i ) , u ( 1 ) = ∑ i = 1 n β i u ( ξ i ) , where ϕ p ( s ) = | s | p − 2 s , p > 1 . We show that it has at least one or two positive solutions under some assumptions by applying the fixed point theorem. The interesting points are that the nonlinear term f is involved with the first-order derivative explicitly and f may change sign.

ON SCHRÖDINGER EQUATIONS WITH INDEFINITE NONLINEARITIES

AbstractIn this paper we consider the problem of finding standing waves – solutions to nonlinear Schrödinger equations with vanishing potential and sign-changing nonlinearities. This involves searching for solutions of the problem (1)We show that the problem has a solution, and the maximum point of the solution is concentrated on a minimum point of some function as ε→0.

Existence of Positive Solutions to Singular -Laplacian General Dirichlet Boundary Value Problems with Sign Changing Nonlinearity

By using the well-known Schauder fixed point theorem and upper and lower solution method, we present some existence criteria for positive solution of an -point singular -Laplacian dynamic equation on time scales with the sign changing nonlinearity. These results are new even for the corresponding differential () and difference equations (), as well as in general time scales setting. As an application, an example is given to illustrate the results.

Triple positive solutions for second-order four-point boundary value problem with sign changing nonlinearities

In this paper, we study the existence of triple positive solutions for second-order four-point boundary value problem with sign changing nonlinearities. We first study the associated Green’s function and obtain some useful properties. Our main tool is the fixed point theorem due to Avery and Peterson. The results of this paper are new and extent previously known 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 .

Solutions form-Point BVP with Sign Changing Nonlinearity

We study the existence of positive solutions for the following nonlinearm-point boundary value problem for an increasing homeomorphism and homomorphism with sign changing nonlinearity:{(ϕ(u′(t)))′+a(t)f(t,u(t))=0,0&lt;t&lt;1,u′(0)=∑i=1m−2aiu′(ξi),u(1)=∑i=1kbiu(ξi)−∑i=k+1sbiu(ξi)−∑i=s+1m−2biu′(ξi), whereϕ:R→Ris an increasing homeomorphism and homomorphism andϕ(0)=0. The nonlinear termfmay change sign. As an application, an example to demonstrate our results is given. The conclusions in this paper essentially extend and improve the known results.

An Approximation Approach to Eigenvalue Intervals for Singular Boundary Value Problems with Sign Changing and Superlinear Nonlinearities

This paper studies the eigenvalue interval for the singular boundary value problem , where may be singular at , , and may change sign and be superlinear at . The approach is based on an approximation method together with the theory of upper and lower solutions.