• Title/Summary/Keyword: Changing sign solutions

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SIGN CHANGING PERIODIC SOLUTIONS OF A NONLINEAR WAVE EQUATION

  • Jung, Tacksun;Choi, Q-Heung
    • Korean Journal of Mathematics
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    • v.16 no.2
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    • pp.243-257
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    • 2008
  • We seek the sign changing periodic solutions of the nonlinear wave equation $u_{tt}-u_{xx}=a(x,t)g(u)$ under Dirichlet boundary and periodic conditions. We show that the problem has at least one solution or two solutions whether $\frac{1}{2}g(u)u-G(u)$ is bounded or not.

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GROUND STATE SIGN-CHANGING SOLUTIONS FOR NONLINEAR SCHRÖDINGER-POISSON SYSTEM WITH INDEFINITE POTENTIALS

  • Yu, Shubin;Zhang, Ziheng
    • Communications of the Korean Mathematical Society
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    • v.37 no.4
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    • pp.1269-1284
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    • 2022
  • This paper is concerned with the following Schrödinger-Poisson system $$\{\begin{array}{lll}-{\Delta}u+V(x)u+K(x){\phi}u=a(x){\mid}u{\mid}^{p-2}u&&\text{ in }{\mathbb{R}}^3,\\-{\Delta}{\phi}=K(x)u^2&&\text{ in }{\mathbb{R}}^3,\end{array}$$ where 4 < p < 6. For the case that K is nonnegative, V and a are indefinite, we prove the above problem possesses one ground state sign-changing solution with exactly two nodal domains by constraint variational method and quantitative deformation lemma. Moreover, we show that the energy of sign-changing solutions is larger than that of the ground state solutions. The novelty of this paper is that the potential a is indefinite and allowed to vanish at infinity. In this sense, we complement the existing results obtained by Batista and Furtado [5].

GROUND STATE SIGN-CHANGING SOLUTIONS FOR A CLASS OF SCHRÖDINGER-POISSON-KIRCHHOFF TYPEPROBLEMS WITH A CRITICAL NONLINEARITY IN ℝ3

  • Qian, Aixia;Zhang, Mingming
    • Journal of the Korean Mathematical Society
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    • v.58 no.5
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    • pp.1181-1209
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    • 2021
  • In the present paper, we are concerned with the existence of ground state sign-changing solutions for the following Schrödinger-Poisson-Kirchhoff system $$\;\{\begin{array}{lll}-(1+b{\normalsize\displaystyle\smashmargin{2}{\int\nolimits_{{\mathbb{R}}^3}}}{\mid}{\nabla}u{\mid}^2dx){\Delta}u+V(x)u+k(x){\phi}u={\lambda}f(x)u+{\mid}u{\mid}^4u,&&\text{in }{\mathbb{R}}^3,\\-{\Delta}{\phi}=k(x)u^2,&&\text{in }{\mathbb{R}}^3,\end{array}$$ where b > 0, V (x), k(x) and f(x) are positive continuous smooth functions; 0 < λ < λ1 and λ1 is the first eigenvalue of the problem -∆u + V(x)u = λf(x)u in H. With the help of the constraint variational method, we obtain that the Schrödinger-Poisson-Kirchhoff type system possesses at least one ground state sign-changing solution for all b > 0 and 0 < λ < λ1. Moreover, we prove that its energy is strictly larger than twice that of the ground state solutions of Nehari type.

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

  • Li, Xiangfeng;Xu, Wanyin
    • Bulletin of the Korean Mathematical Society
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    • v.47 no.2
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    • pp.411-422
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    • 2010
  • 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 $(\varphi_p(u'))'+a(t)f(t,u(t),u'(t))=0$, subject to some boundary conditions. As an application, we also give an example to illustrate our results.

Positive Solutions of Nonlinear Neumann Boundary Value Problems with Sign-Changing Green's Function

  • Elsanosi, Mohammed Elnagi M.
    • Kyungpook Mathematical Journal
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    • v.59 no.1
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    • pp.65-71
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    • 2019
  • This paper is concerned with the existence of positive solutions of the nonlinear Neumann boundary value problems $$\{u^{{\prime}{\prime}}+a(t)u={\lambda}b(t)f(u),\;t{\in}(0,1),\\u^{\prime}(0)=u^{\prime}(1)=0$$, where $a,b{\in}C[0,1]$ with $a(t)>0,\;b(t){\geq}0$ and the Green's function of the linear problem $$\{u^{{\prime}{\prime}}+a(t)u=0,\;t{\in}(0,1),\\u^{\prime}(0)=u^{\prime}(1)=0$$ may change its sign on $[0,1]{\times}[0,1]$. Our analysis relies on the Leray-Schauder fixed point theorem.

TRIPLE SOLUTIONS FOR THREE-ORDER PERIODIC BOUNDARY VALUE PROBLEMS WITH SIGN CHANGING NONLINEARITY

  • Tan, Huixuan;Feng, Hanying;Feng, Xingfang;Du, Yatao
    • Journal of applied mathematics & informatics
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    • v.32 no.1_2
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    • pp.75-82
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    • 2014
  • In this paper, we consider the periodic boundary value problem with sign changing nonlinearity $$u^{{\prime}{\prime}{\prime}}+{\rho}^3u=f(t,u),\;t{\in}[0,2{\pi}]$$, subject to the boundary value conditions: $$u^{(i)}(0)=u^{(i)}(2{\pi}),\;i=0,1,2$$, where ${\rho}{\in}(o,{\frac{1}{\sqrt{3}}})$ is a positive constant and f(t, u) is a continuous function. Using Leggett-Williams fixed point theorem, we provide sufficient conditions for the existence of at least three positive solutions to the above boundary value problem. The interesting point is the nonlinear term f may change sign.

POSITIVE SOLUTIONS FOR NONLINEAR m-POINT BVP WITH SIGN CHANGING NONLINEARITY ON TIME SCALES

  • HAN, WEI;REN, DENGYUN
    • Journal of applied mathematics & informatics
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    • v.35 no.5_6
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    • pp.551-563
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    • 2017
  • In this paper, by using fixed point theorems in cones, the existence of positive solutions is considered for nonlinear m-point boundary value problem for the following second-order dynamic equations on time scales $$u^{{\Delta}{\nabla}}(t)+a(t)f(t,u(t))=0,\;t{\in}(0,T),\;{\beta}u(0)-{\gamma}u^{\Delta}(0)=0,\;u(T)={\sum_{i=1}^{m-2}}\;a_iu({\xi}_i),\;m{\geq}3$$, where $a(t){\in}C_{ld}((0,T),\;[0,+{\infty}))$, $f{\in}C([0,T]{\times}[0,+{\infty}),\;(-{\infty},+{\infty}))$, the nonlinear term f is allowed to change sign. We obtain several existence theorems of positive solutions for the above boundary value problems. In particular, our criteria generalize and improve some known results [15] and the obtained conditions are different from related literature [14]. As an application, an example to demonstrate our results is given.

THE EXISTENCE OF TWO POSITIVE SOLUTIONS FOR $m$-POINT BOUNDARY VALUE PROBLEM WITH SIGN CHANGING NONLINEARITY

  • Liu, Jian
    • Journal of applied mathematics & informatics
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    • v.30 no.3_4
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    • pp.517-529
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    • 2012
  • In this paper, the existence theorem of two positive solutions is established for nonlinear m-point boundary value problem by using an inequality for the following third-order differential equations $$({\phi}(u^{\prime\prime}))^{\prime}+a(t)f(t,u(t))=0,\;t{\in}(0,1)$$, $${\phi}(u^{\prime\prime}(0))=\sum^{m-2}_{i=1}a_i{\phi}(u^{\prime\prime}({\xi}_i)),\;u^{\prime}(1)=0,\;u(0)=\sum^{m-2}_{i=1}b_iu({\xi}_i)$$, where ${\phi}:R{\rightarrow}R$ is an increasing homeomorphism and homomorphism and $\phi(0)=0$. The nonlinear term f may change sign, as an application, an example to demonstrate our results is given.

ASYMPTOTIC BEHAVIOR OF SOLUTIONS OF FORCED NONLINEAR NEUTRAL DIFFERENCE EQUATIONS

  • Liu, Yuji;Ge, Weigao
    • Journal of applied mathematics & informatics
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    • v.16 no.1_2
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    • pp.37-51
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    • 2004
  • In this paper, we consider the asymptotic behavior of solutions of the forced nonlinear neutral difference equation $\Delta[x(n)-\sumpi(n)x(n-k_i)]+\sumqj(n)f(x(n-\iota_j))=r(n)$ with sign changing coefficients. Some sufficient conditions for every solution of (*) to tend to zero are established. The results extend and improve some known theorems in literature.

ASYMPTOTIC STABILIZATION FOR A DISPERSIVE-DISSIPATIVE EQUATION WITH TIME-DEPENDENT DAMPING TERMS

  • Yi, Su-Cheol
    • Journal of the Chungcheong Mathematical Society
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    • v.33 no.4
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    • pp.445-468
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    • 2020
  • A long-time behavior of global solutions for a dispersive-dissipative equation with time-dependent damping terms is investigated under null Dirichlet boundary condition. By virtue of an appropriate new Lyapunov function and the Lojasiewicz-Simon inequality, we show that any global bounded solution converges to a steady state and get the rate of convergence as well, when damping coefficients are integrally positive and positive-negative, respectively. Moreover, under the assumptions on on-off or sign-changing damping, we derive an asymptotic stability of solutions.