Weighted polyharmonic equation with Navier boundary conditions in a half space

We study positive solutions of the following polyharmonic equation with Hardy weights associated to Navier boundary conditions on a half space:?????(-?)~mu(x)=u~p(x)/|x|~s,in R_+~n,u(x)=-?u(x)=…=(-?)~(m-1)u(x)=0,on ?R_+~n,(0.1)where m is any positive integer satisfying 02mn.We first prove that the positive solutions of(0.1)are super polyharmonic,i.e.,(-?)~iu0,i=0,1,...,m-1.(0.2) For α=2m,applying this important property,we establish the equivalence between (0.1) and the integral equation u(x)=c_n∫R_+~n(1/|x-y|~(n-α)-1/|x~*-y|~(n-α))u~p(y)/|y|~sdy,(0.3) where x~*=(x1,...,x_(n-1),-x_n) is the reflection of the point x about the plane R~(n-1).Then,we use the method of moving planes in integral forms to derive rotational symmetry and monotonicity for the positive solution of(0.3),in whichαcan be any real number between 0 and n.By some Pohozaev type identities in integral forms,we prove a Liouville type theorem—the non-existence of positive solutions for(0.1).     

Infinitely many solutions for a nonlinear Schrödinger equation with general nonlinearity

We prove the existence of infinitely many solutions for

$$\begin{aligned} - \Delta u + V(x) u = f(u) \quad \text { in } \mathbb {R}^N, \quad u \in H^1(\mathbb {R}^N), \end{aligned}$$

where V(x) satisfies \(\lim _{|x| \rightarrow \infty } V(x) = V_\infty >0\) and some conditions. We require conditions on f(u) only around 0 and at \(\infty \).

     

Dynamic control for non-linear systems with delay

We consider some class of non-linear systems of the form

$\dot x = A( \cdot )x + \sum\limits_{i = 1}^l {A_i ( \cdot )x(t - \tau _i (t)) + b( \cdot )u} ,$

where A(·) ∈ ? ^{n × n} , A _i (·) ∈ ? ^{n × n} , b(·) ∈ ? ⁿ , whose coefficients are arbitrary uniformly bounded functionals.

A special type of the Lyapunov-Krasovskii functional is used to synthesize dynamic control described by the equation

$\dot u = \rho ( \cdot )u + (m( \cdot ),x),$

where ρ(·) ∈ ?¹, m(·) ∈ ? ⁿ , which makes the system globally asymptotically stable. Also, the situation is considered where the control u enters into the system not directly but through a pulse element performing an amplitude-frequency modulation.

     

Infinitely many solutions for Kirchhoff equations with sign-changing potential and Hartree nonlinearity

This paper is concerned with the following Kirchhoff-type equations:

$$\begin{aligned} \left\{ \begin{array}{ll} \displaystyle -\big (a+b\int _{\mathbb {R}^{3}}|\nabla u|^{2}\mathrm {d}x\big )\Delta u+ V(x)u+\mu \phi |u|^{p-2}u=f(x, u)+g(x,u), &{} \text{ in } \mathbb {R}^{3},\\ (-\Delta )^{\frac{\alpha }{2}} \phi = \mu |u|^{p}, &{} \text{ in } \mathbb {R}^{3},\\ \end{array} \right. \end{aligned}$$

where \(a>0,~b,~\mu \ge 0\) are constants, \(\alpha \in (0,3)\), \(p\in [2,3+2\alpha )\), the potential V(x) may be unbounded from below and \(\phi |u|^{p-2}u\) is a Hartree-type nonlinearity. Under some mild conditions on V(x), f(x, u) and g(x, u), we prove that the above system has infinitely many nontrivial solutions. Specially, our results cover the general Schrödinger equations, the Kirchhoff equations and the Schrödinger–Poisson system.

     

Positive solutions of nonlinear three-point boundary-value problems

Let a∈C[0,1], b∈C([0,1],(−∞,0]). Let φ₁(t) be the unique solution of the linear boundary value problem

u″(t)+a(t)u′(t)+b(t)u(t)=0,t∈(0,1),u(0)=0,u(1)=1.     

On correspondence between solutions of a family of cubic Thue equations and isomorphism classes of the simplest cubic fields

Let m?−1 be an integer. We give a correspondence between integer solutions to the parametric family of cubic Thue equations

X³−mX²Y−(m+3)XY²−Y³=λ     

Existence results of a m-point boundary value problem at resonance

Let ξ_i∈(0,1), a_i∈(0,∞), i=1,…,m−2, be given constants satisfying ∑^m−2_i=1a_i=1 and 0<ξ₁<ξ₂<?<ξ_m−2<1. We show the existence of solutions for the m-point boundary value problem

x″=f(t,x,x′),t∈(0,1),     

Least energy solutions of nonlinear Schrödinger equations involving the fractional Laplacian and potential wells

We are concerned with the existence of least energy solutions of nonlinear Schrödinger equations involving the fractional Laplacian

$$\begin{array}{*{20}c} {( - \Delta )^s u(x) + \lambda V(x)u(x) = u(x)^{p - 1} ,} & {u(x) \geqslant 0,} & {x \in \mathbb{R}^N ,} \\ \end{array} $$

for sufficiently large λ, 2 < p < \(\frac{{2N}}{{N - 2s}}\) for N ≥ 2. V (x) is a real continuous function on R^N. Using variational methods we prove the existence of least energy solution u _λ(x) which localizes near the potential well int V ^?1(0) for λ large. Moreover, if the zero sets int V ?1(0) of V (x) include more than one isolated component, then u _λ(x) will be trapped around all the isolated components. However, in Laplacian case s = 1, when the parameter λ is large, the corresponding least energy solution will be trapped around only one isolated component and become arbitrarily small in other components of int V ^?1(0). This is the essential difference with the Laplacian problems since the operator (?Δ)^s is nonlocal.

     

Optimization of the control by elastic boundary forces at two ends of a string in an arbitrarily large time interval

We further develop the method, devised earlier by the authors, which permits finding closed-form expressions for the optimal controls by elastic boundary forces applied at two ends, x = 0 and x = l, of a string. In a sufficiently large time T, the controls should take the string vibration process, described by a generalized solution u(x, t) of the wave equation

$$u_{tt} (x,t) - u_{tt} (x,t) = 0,$$

from an arbitrary initial state

$$\{ u(x,0) = \varphi (x), u_t (x,0) = \psi (x)$$

to an arbitrary terminal state

$$\{ u(x,T) = \hat \varphi (x), u_t (x,T) = \hat \psi (x).$$

     

A radial symmetry and Liouville theorem for systems involving fractional Laplacian

We investigate the nonnegative solutions of the system involving the fractional Laplacian:

$$\left\{ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {( - \Delta )^\alpha u_i (x) = f_i (u),} & {x \in \mathbb{R}^n , i = 1,2, \ldots ,m,} \\ \end{array} } \\ {u(x) = (u_1 (x),u_2 (x), \ldots ,u_m (x)),} \\ \end{array} } \right.$$

where 0 < α < 1, n > 2, f _i (u), 1 ≤ i ≤ m, are real-valued nonnegative functions of homogeneous degree p _i ≥ 0 and nondecreasing with respect to the independent variables u ₁, u ₂,..., u _m . By the method of moving planes, we show that under the above conditions, all the positive solutions are radially symmetric and monotone decreasing about some point x ₀ if p _i = (n + 2α)/(n ? 2α) for each 1 ≤ i ≤ m; and the only nonnegative solution of this system is u ≡ 0 if 1 < p _i < (n + 2α)/(n ? 2α) for all 1 ≤ i ≤ m.

     

A positive solution of a nonlinear Schrdinger system with nonconstant potentials

In this paper, we study the existence of positive solutions to the following Schr¨odinger system:{-?u + V_1(x)u = μ_1(x)u~3+ β(x)v~2u, x ∈R~N,-?v + V_2(x)v = μ_2(x)v~3+ β(x)u~2v, x ∈R~N,u, v ∈H~1(R~N),where N = 1, 2, 3; V_1(x) and V_2(x) are positive and continuous, but may not be well-shaped; and μ_1(x), μ_2(x)and β(x) are continuous, but may not be positive or anti-well-shaped. We prove that the system has a positive solution when the coefficients Vi(x), μ_i(x)(i = 1, 2) and β(x) satisfy some additional conditions.     

Small into isomorphisms on uniformly smooth spaces

Let X be a uniformly smooth infinite dimensional Banach space, and (Ω,Σ,μ) be a σ-finite measure space. Suppose that T:X→L∞(Ω,Σ,μ) satisfies

(1−ε)‖x‖?‖Tx‖?‖x‖,∀x∈X,     

Ground state solutions of Nehari–Pohozaev type for Kirchhoff-type problems with general potentials

This paper is dedicated to studying the following Kirchhoff-type problem

$$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} -\left( a+b\int _{\mathbb {R}^3}|\nabla u|^2\mathrm {d}x\right) \triangle u+V(x)u=f(u), &{} x\in \mathbb {R}^3; \\ u\in H^1(\mathbb {R}^3), \end{array} \right. \end{aligned}$$

(0.1)

where \(a>0,\,b\ge 0\) are two constants, V(x) is differentiable and \(f\in \mathcal {C}(\mathbb {R}, \mathbb {R})\). By introducing some new tricks, we prove that the above problem admits a ground state solution of Nehari–Pohozaev type and a least energy solution under some mild assumptions on V and f. Our results generalize and improve the ones in Guo (J Differ Equ 259:2884–2902, 2015) and Li and Ye (J Differ Equ 257:566–600, 2014) and some other related literature.

     

Linear Approximation and Asymptotic Expansion of Solutions in Many Small Parameters for a Nonlinear Kirchhoff Wave Equation with Mixed Nonhomogeneous Conditions

In this paper, we consider the following nonlinear Kirchhoff wave equation

$\left\{\begin{array}{l}u_{tt}-\frac{\partial }{\partial x}(\mu (u,\Vert u_{x}\Vert ^{2})u_{x})=f(x,t,u,u_{x},u_{t}),\quad 0

(1)

where \(\widetilde{u}_{0}\), \(\widetilde{u}_{1}\), μ, f, g are given functions and \(\Vert u_{x}\Vert ^{2}=\int_{0}^{1}u_{x}^{2}(x,t)dx.\) To the problem (1), we associate a linear recursive scheme for which the existence of a local and unique weak solution is proved by applying the Faedo–Galerkin method and the weak compact method. In particular, motivated by the asymptotic expansion of a weak solution in only one, two or three small parameters in the researches before now, an asymptotic expansion of a weak solution in many small parameters appeared on both sides of (1)₁ is studied.

     

A New Result on Multiplicity of Nontrivial Solutions for the Nonhomogenous Schrödinger–Kirchhoff Type Problem in <Emphasis Type="Italic">R</Emphasis><Superscript><Emphasis Type="Italic">N</Emphasis></Superscript>

In this paper, we consider the following nonhomogenous Schrödinger–Kirchhoff type problem

$$\left\{ \begin{array}{ll} - (a+b\int_{R^{N}}|\nabla u|^{2}dx)\triangle u + V(x)u =f(x,u)+g(x), & \,\,\,{\rm for} \, x \in R^N, \\ u(x)\rightarrow0, & \,\, {\rm as}\, |x|\rightarrow\infty,\end{array}\right.$$

(0.1)

where constants a > 0, b ≥ 0, N = 1, 2 or 3, \({V\in C(R^{N},R)}\), \({f\in C(R^{N} \times R, R)}\) and \({g\in L^{2}(R^{N})}\). Under more relaxed assumptions on the nonlinear term f that are much weaker than those in Chen and Li (Nonlinear Anal RWA 14:1477–1486, 2013), using some new proof techniques especially the verification of the boundedness of Palais–Smale sequence, a new result on multiplicity of nontrivial solutions for the problem (1.1) is obtained, which sharply improves the known result of Theorem 1.1 in Chen and Li (Nonlinear Anal RWA 14:1477–1486, 2013).

     

Limit problems for a Fractional p-La placian as $${ p\to\infty}$$

The purpose of this work is the analysis of the solutions to the following problems related to the fractional p-Laplacian in a Lipschitzian bounded domain \({\Omega \subset \mathbb{R}^N}\),

$$\left\{\begin{array}{lll}-\int_{\mathbb{R}^N}\frac{|u(y)-u(x)|^{p-2}(u(y)-u(x))}{|x-y|^{\alpha p}}\;dy=f(x,u)\;\;&x\in \Omega,\\ u=g(x) &x\in\mathbb{R}^N\setminus \Omega,\end{array}\right.$$

where \({\alpha\in(0,1)}\) and the exponent p goes to infinity. In particular we will analyze the cases:

1. (i)
   \({f=f(x).}\)
    
2. (ii)
   \({f=f(u)=|u|^{\theta(p)-1} u \, {\rm with} \, 0 < \theta(p) < p -1 \, {\rm and} \, \lim_{p\to\infty}\frac{\theta(p)}{p-1}=\Theta < 1 \, {\rm with} \, g \geq 0.}\)
    

We show the convergence of the solutions to certain limit as \({p\to\infty}\) and identify the limit equation. In both cases, the limit problem is closely related to the Infinity Fractional Laplacian:

$$\mathcal{L}_\infty v(x)=\mathcal{L}_\infty^+ v(x)+\mathcal{L}_\infty^- v(x),$$

where

$$\mathcal{L}_\infty^+ v(x)=\sup_{y\in\mathbb{R}^N}\frac{v(y)-v(x)}{|y-x|^\alpha}, \quad \mathcal{L}_\infty^- v(x)=\inf_{y\in\mathbb{R}^N}\frac{v(y)-v(x)}{|y-x|^\alpha}.$$

     

Periodic solutions of a semilinear elliptic equation with a fractional Laplacian

We consider periodic solutions of the following problem associated with the fractional Laplacian

$$(-\partial _{xx})^s u(x) + F'(u(x))=0,\quad u(x)=u(x+T),\quad \text{ in } \, \mathbb {R}, $$

where \((-\partial _{xx})^s\) denotes the usual fractional Laplace operator with \(0<s<1\). The primitive function F of the nonlinear term is a smooth double-well potential. We prove the existence of periodic solutions with large period T using variational methods. An estimate of the energy of the periodic solutions is also established.

     

Solutions of perturbed Schrödinger equations with critical nonlinearity

We consider the perturbed Schrödinger equation

$\left\{\begin{array}{ll}{- \varepsilon ^2 \Delta u + V(x)u = P(x)|u|^{p - 2} u + k(x)|u|^{2* - 2} u} &; {\text{for}}\, x \in {\mathbb{R}}^N\\ \qquad \qquad \quad {u(x) \rightarrow 0} &; \text{as}\, {|x| \rightarrow \infty} \end{array} \right.$

where \(N\geq 3, \ 2^*=2N/(N-2)\) is the Sobolev critical exponent, \(p\in (2, 2^*)\) , P(x) and K(x) are bounded positive functions. Under proper conditions on V we show that it has at least one positive solution provided that \(\varepsilon\leq{\mathcal{E}}\) ; for any \(m\in{\mathbb{N}}\) , it has m pairs of solutions if \(\varepsilon\leq{\mathcal{E}}_{m}\) ; and suppose there exists an orthogonal involution \(\tau:{\mathbb{R}}^{N}\to{\mathbb{R}}^{N}\) such that V(x), P(x) and K(x) are τ -invariant, then it has at least one pair of solutions which change sign exactly once provided that \(\varepsilon\leq{\mathcal{E}}\) , where \({\mathcal{E}}\) and \({\mathcal{E}}_{m}\) are sufficiently small positive numbers. Moreover, these solutions \(u_\varepsilon\to 0\) in \(H^1({\mathbb{R}}^N)\) as \(\varepsilon\to 0\) .

     

A boundary blow-up for a class of quasilinear elliptic problems with a gradient term

By a perturbation method and constructing comparison functions, we show the exact asymptotic behavior of solutions near the boundary to quasilinear elliptic problem $$\left\{\begin{array}{ll}\mbox{div}\left(|\nabla u|^{m-2}\nabla u\right)-|\nabla u(x)|^{q(m-1)}=b(x)g(u),\quad x\in \Omega,\\u>0,\quad x\in \Omega,\\u|_{\partial\Omega}=+\infty,\end{array}\right.$$ where Ω is a C ² bounded domain with smooth boundary, m>1,q∈(1,m/(m?1)], g∈C[0,∞)∩C ¹(0,∞), g(0)=0, g is increasing on [0,∞), and b is non-negative and non-trivial in Ω, which may be singular on the boundary.     

Equilibria of a nonautonomous Lotka-Volterra model with shelter for the prey

For the nonautonomous Lotka-Volterra model

$\dot x = \alpha (t)(x - M^{ - 1} x^2 - K^{ - 1} (x - \phi (x,y))y),\dot y = \beta (t)y(L^{ - 1} (x - \phi (x,y)) - 1),$

where some part φ(x, y) of the prey population is out of reach of the predator, we obtain sufficient conditions for the existence of a positive asymptotically stable equilibrium in the domain of admissible values of the variables x and y. We consider the cases in which φ(x, y) = m, φ(x, y) = mx, and φ(x, y) = my.