Nehari Type Research Articles

Hille–Nehari type criteria and conditionally oscillatory half-linear differential equations

Hille–Nehari type criteria and conditionally oscillatory half-linear differential equations

Ground state solutions of Pohoz̆aev type and Nehari type for a class of nonlinear Choquard equations

In this paper, we study the autonomous Choquard equation−Δu+u=(Iα⁎F(u))f(u),inRN, where N≥3, 0<α<N, Iα is a Riesz potential, and f∈C(R,R) satisfies the general Berestycki–Lions conditions. In Sec. 2, combining constrained variational method with deformation lemma, we obtain a ground state solution of Pohoz̆aev type for the above equation. In Sec. 3, using non-Nehari manifold method, we prove that the above equation has a ground state solution of Nehari type. The results improve some ones in [12].

Ground state sign-changing solutions for semilinear Dirichlet problems

In the present paper, we consider the existence of ground state sign-changing solutions for the semilinear Dirichlet problem \t\t\t0.1{−△u+λu=f(x,u),x∈Ω;u=0,x∈∂Ω,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\left \\{ \\textstyle\\begin{array}{l@{\\quad}l} -\\triangle u+\\lambda u=f(x, u), & \\hbox{$x\\in\\Omega$;} \\\\ u=0, & \\hbox{$x\\in\\partial\\Omega$,} \\end{array}\\displaystyle \\right . $$\\end{document} where Omegasubsetmathbb{R}^{N} is a bounded domain with a smooth boundary ∂Ω, lambda>-lambda_{1} is a constant, lambda_{1} is the first eigenvalue of (-triangle, H_{0}^{1}(Omega)), and fin C(Omegatimesmathbb{R}, mathbb{R}). Under some standard growth assumptions on f and a weak version of Nehari type monotonicity condition that the function tmapsto f(x, t)/|t| is non-decreasing on (-infty, 0)cup(0, infty) for every xinOmega, we prove that (0.1) possesses one ground state sign-changing solution, which has precisely two nodal domains. Our results improve and generalize some existing ones.

Sign-changing solutions for non-local elliptic equations involving the fractional Laplacain

In this paper, we consider the existence of sign-changing solutions for fractional elliptic equations of the form \begin{equation*} \left\{\begin{array}{ll} (-\Delta)^s u=f(x,u) & \text{in}\ \Omega , \\ u=0 & \text{in}\ \mathbb R^N\setminus \Omega, \end{array} \right. \end{equation*} where $s\in(0,1)$ and $\Omega\subset \mathbb R^N$ is a bounded smooth domain. Since the non-local operator $(-\Delta)^s$ is involved in the equation, the variational functional of the equation has totally different properties from the local cases. By introducing some new ideas, we prove, via variational method and the method of invariant sets of descending flow, that the problem has a positive solution, a negative solution and a sign-changing solution under suitable conditions. Moreover, if $f(x,u)$ satisfies a monotonicity condition, we show that the problem has a least energy sign-changing solution with its energy is strictly larger than that of the ground state solution of Nehari type. We also obtain an unbounded sequence of sign-changing solutions if $f(x,u)$ is odd in $u$.

Ground state sign-changing solutions for a class of generalized quasilinear Schrödinger equations with a Kirchhoff-type perturbation

We investigate the existence of ground state sign-changing solutions for the following elliptic equation of Kirchhoff type: $$\begin{aligned} \begin{aligned}&\left( 1+b\int _{{{\mathbb {R}}}^3}g^2(u)|\nabla u|^2\mathrm{d}x\right) \left[ -\text {div}(g^2(u)\nabla u)+g(u)g'(u)|\nabla u|^2\right] \\&\quad +V(x)u=K(x)f(u), \end{aligned} \end{aligned}$$ where \(x\in {{\mathbb {R}}}^3, b>0, g\in \mathcal {C}^1({{\mathbb {R}}},{{\mathbb {R}}}^{+}), V(x)\), and K(x) are both positive continuous functions. First, using some new analytical techniques and non-Nehari manifold method, we obtain one ground state sign-changing solution \(v_b=G^{-1}(u_b)\). Moreover, we prove that the energy of \(v_b=G^{-1}(u_b)\) is strictly larger than twice that of the ground state solutions of Nehari type. We also establish the convergence property of \(v_b=G^{-1}(u_b)\) as the parameter \(b\searrow 0\). Our results improve and generalize some results in Li et al. (J Math Anal Appl 443:11–38, 2016).

Oscillation criteria for third-order functional half-linear dynamic equations

In this paper, we study the third-order functional dynamic equation {r2(t)ϕα2([r1(t)ϕα1(xΔ(t))]Δ)}Δ+q(t)ϕα(x(g(t)))=0,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\bigl\\{ r_{2}(t)\\phi_{\\alpha_{2}} \\bigl( \\bigl[ r_{1}(t) \\phi _{\\alpha _{1}} \\bigl( x^{\\Delta}(t) \\bigr) \\bigr] ^{\\Delta} \\bigr) \\bigr\\} ^{\\Delta}+q(t)\\phi_{\\alpha} \\bigl( x\\bigl(g(t)\\bigr) \\bigr) =0, $$\\end{document} on an upper-unbounded time scale mathbb{T}. We will extend the so-called Hille and Nehari type criteria to third-order dynamic equations on time scales. This work extends and improves some known results in the literature on third-order nonlinear dynamic equations and the results are established for a time scale mathbb{T} without assuming certain restrictive conditions on mathbb{T}. Some examples are given to illustrate the main results.

Existence of ground state sign‐changing solutions for p‐Laplacian equations of Kirchhoff type

In this paper, we prove the existence of ground state sign‐changing solutions for the following class of elliptic equation: urn:x-wiley:mma:media:mma4370:mma4370-math-0001 where , and K(x) are positive continuous functions. Firstly, we obtain one ground state sign‐changing solution ub by using some new analytical skills and non‐Nehari manifold method. Furthermore, the energy of ub is strictly larger than twice that of the ground state solutions of Nehari type. We also establish the convergence property of ub as the parameter b↘0. Copyright © 2017 John Wiley &amp; Sons, Ltd.

Asymptotic behavior of third-order differential equations with nonpositive neutral coefficients and distributed deviating arguments

Using the Riccati transformation technique, we present several sufficient conditions that guarantee that all solutions to a third-order differential equation with nonpositive neutral coefficients and distributed deviating arguments are either oscillatory or converge to zero asymptotically. In particular, we establish Hille and Nehari type criteria. Two examples are given to demonstrate the practicability of the main results.

Ground states for diffusion system with periodic and asymptotically periodic nonlinearity

In this paper, we study the following diffusion system {∂tu−Δxu+b(t,x)⋅∇xu+V(x)u=g(t,x,v),−∂tv−Δxv−b(t,x)⋅∇xv+V(x)v=f(t,x,u) where b∈C1(R×RN,RN) and V(x)∈C(RN,R). By using non-Nehari manifold method developed in Tang (2014), we investigate the existence of the ground state solutions of Nehari type for the above problem with periodic and asymptotically periodic nonlinearity.

Solutions to a nonlinear Schrödinger equation with periodic potential and zero on the boundary of the spectrum

We study the following nonlinear Schrodinger equation\begin{equation*}\begin{cases} -\Delta u + V(x) u = g(x,u) &amp; \hbox{for } x\in\R^N,\\ u(x)\to 0 &amp; \hbox{as } |x|\to\infty,\end{cases}\end{equation*}where $V\colon \R^N\to\R$ and $g\colon \R^N\times\R\to\R$ are periodic in $x$. We assume that $0$ is a right boundary point of the essential spectrum of $-\Delta+V$. The superlinear and subcritical term g satisfies a Nehari type monotonicity condition. We employ a Nehari manifold type technique in a strongly indefitnite setting and obtain the existence of a ground state solution. Moreover, we get infinitely many geometrically distinct solutions provided that $g$ is odd.

Positive solutions with a complex behavior for superlinear indefinite ODEs on the real line

We show the existence of infinitely many positive solutions, defined on the real line, for the nonlinear scalar ODEu¨+(a+(t)−μa−(t))u3=0, where a is a periodic, sign-changing function, and the parameter μ>0 is large. Such solutions are characterized by the fact of being either small or large in each interval of positivity of a. In this way, we find periodic solutions, having minimal period arbitrarily large, and bounded non-periodic solutions, exhibiting a complex behavior. The proof is variational, exploiting suitable natural constraints of Nehari type.

Hille and Nehari type oscillation criteria for higher order dynamic equations on time scales

Hille and Nehari type oscillation criteria for higher order dynamic equations on time scales

Non-Nehari manifold method for asymptotically periodic Schrödinger equations

We consider the semilinear Schrödinger equation $$\left\{ {\begin{array}{*{20}c} { - \Delta u + V(x)u = f(x,u), \mathbb{R}^N ,} \\ {u \in H^1 (\mathbb{R}^N ),} \\ \end{array} } \right.$$ where f is a superlinear, subcritical nonlinearity. We mainly study the case where V(x) = V 0(x) + V 1(x), V 0 ∈ C(ℝN), V 0(x) is 1-periodic in each of x 1, x 2, …, x N and sup[σ(−Δ + V 0) ∩ (−∞, 0)] < 0 < inf[σ(−Δ + V 0) ∩ (0, ∞)], V 1 ∈ C(ℝN) and lim|x|→∞ V 1(x) = 0. Inspired by previous work of Li et al. (2006), Pankov (2005) and Szulkin and Weth (2009), we develop a more direct approach to generalize the main result of Szulkin and Weth (2009) by removing the “strictly increasing” condition in the Nehari type assumption on f(x, t)/|t|. Unlike the Nahari manifold method, the main idea of our approach lies on finding a minimizing Cerami sequence for the energy functional outside the Nehari-Pankov manifold N 0 by using the diagonal method.

On the least energy sign-changing solutions for a nonlinear elliptic system

In this paper, as bound state solutions we consider least energy sign-changing solutions to a nonlinear elliptic systemwhich consists of N-equations defined on a bounded domain $\Omega$.For any subset $K\subset \{1,\cdots, N\}$,we show the existence of sign-changing solution$\vec{u}=(u_1,\cdots,u_n)$ such that,for $i\in K$, $u_i$ are sign-changing functions that change sign exactly oncein $\Omega$,and, for $i\notin K$, $u_i$ are one sign functions.We give a variational characterization of such solutions on modified Nehari type constrained sets.

NON-NEHARI MANIFOLD METHOD FOR SUPERLINEAR SCHRÖDINGER EQUATION

We consider the boundary value problem \begin{equation} \tag{0.1} \left\{ \begin{array}{ll} -\triangle u+V(x)u=f(x, u), \ \ \ \ & x\in \Omega,\\ u=0, \ \ \ \ & x\in \partial\Omega, \end{array} \right. \end{equation} where $ \Omega \subset \mathbb R^N$ be a bounded domain, $\inf_{\Omega}V(x)\gt -\infty$, $f$ is a superlinear, subcritical nonlinearity. Inspired by previous work of Szulkin and Weth (2009)) [21] and (2010) [22], we develop a more direct and simpler approach on the basis of one used in [21], to deduce weaker conditions under which problem (0.1) has a ground state solution of generalized Nehari type or infinity many nontrivial solutions. Unlike the Nehari manifold method, the main idea of our approach lies on finding a minimizing Cerami sequence for the energy functional outside the generalized Nehari manifold by using the diagonal method.

On a nonlinear Schrödinger equation with indefinite potential

We investigate the semilinear Schrödinger equation −Δu+V(x)u=Q(x)f(u),inRN, where N≥3, V(x)∈LlocN/2(RN) and Q(x)∈L∞(RN), V(x) and Q(x) respectively tend to some positive limits V∞ and Q∞ as |x|→∞, and f∈C(R) is a superlinear and subcritical function. Assuming some weak one-sided asymptotic estimates for V(x) and Q(x), we prove that the above equation has a positive ground state solution and a least energy nodal solution via the minimization method constrained to Nehari type sets.

Hille and Nehari type criteria for third-order delay dynamic equations†

The objective of this note is to present new Hille and Nehari type asymptotic criteria for a class of third-order delay dynamic equations on a time scale. Assumptions in our theorems are less restrictive, whereas the proofs are significantly simpler compared to those reported in the literature. The results obtained extend and improve some previous results.

Ground state solutions for nonperiodic Dirac equation with superquadratic nonlinearity

This paper is concerned with the following nonperiodic Dirac equation \documentclass[12pt]{minimal}\begin{document}$-i\sum _{k=1}^{3}\alpha _{k}\partial _{k}u + a\beta u + M(x)u = F_{u}(x, u)$\end{document}−i∑k=13αk∂ku+aβu+M(x)u=Fu(x,u) in \documentclass[12pt]{minimal}\begin{document}$\mathbb {R}^3$\end{document}R3, where M(x) is a vector potential. Under weak superquadratic condition on the nonlinearity, we establish the existence of ground state solutions of Nehari type by using the generalized Nehari manifold method developed recently by Szulkin and Weth for above problem.

Some oscillation criteria for the second-order linear delay differential equation

Some Wintner and Nehari type oscillation criteria are established for the second-order linear delay differential equation.

Existence of Standing Waves for Coupled Nonlinear Schrödinger Equations

In this paper we study the existence of standing waves for coupled nonlinear Schrödinger equations. The interaction between equations plays an important role in our study. When the interaction is strong, the least energy solution is a solution whose both components are positive. When the interaction is weak, the least energy solution is a semitrivial solution, namely a solution of a form $(u_1,0)$ or $(0,u_2)$. Moreover, minimizing method on the Nehari type manifold with codimension 2 gives us a positive solution when the interaction is weak.