Volume 21, Issue 3

In this paper, we investigate the problem of the existence and multiplicity of periodic solutions to the planar Hamiltonian system x′=-λ⁢α⁢(t)⁢f⁢(y)x^{\prime}=-\lambda\alpha(t)f(y), y′=λ⁢β⁢(t)⁢g⁢(x)y^{\prime}=\lambda\beta(t)g(x), where α,β\alpha,\beta are non-negative 𝑇-periodic coefficients and λ>0\lambda>0. We focus our study to the so-called “degenerate” situation, namely when the set Z:=supp⁡α∩supp⁡βZ:=\operatorname{supp}\alpha\cap\operatorname{supp}\beta has Lebesgue measure zero. It is known that, in this case, for some choices of 𝛼 and 𝛽, no nontrivial 𝑇-periodic solution exists. On the opposite, we show that, depending of some geometric configurations of 𝛼 and 𝛽, the existence of a large number of 𝑇-periodic solutions (as well as subharmonic solutions) is guaranteed (for λ>0\lambda>0 and large). Our proof is based on the Poincaré–Birkhoff twist theorem. Applications are given to Volterra’s predator-prey model with seasonal effects.

In this paper, we study the following nonlinear magnetic Kirchhoff equation: { - ( a ⁢ ϵ 2 + b ⁢ ϵ ⁢ [ u ] A / ϵ 2 ) ⁢ Δ A / ϵ ⁢ u + V ⁢ ( x ) ⁢ u = f ⁢ ( | u | 2 ) ⁢ u in  ⁢ ℝ 3 , u ∈ H 1 ⁢ ( ℝ 3 , ℂ ) , \left\{\begin{aligned} &\displaystyle{-}(a\epsilon^{2}+b\epsilon[u]_{A/% \epsilon}^{2})\Delta_{A/\epsilon}u+V(x)u=f(\lvert u\rvert^{2})u&&\displaystyle% \phantom{}\text{in }\mathbb{R}^{3},\\ &\displaystyle u\in H^{1}(\mathbb{R}^{3},\mathbb{C}),\end{aligned}\right. where ϵ>0{\epsilon>0}, a,b>0{a,b>0} are constants, V:ℝ3→ℝ{V:\mathbb{R}^{3}\rightarrow\mathbb{R}} and A:ℝ3→ℝ3{A:\mathbb{R}^{3}\rightarrow\mathbb{R}^{3}} are continuous potentials, and ΔA⁢u{\Delta_{A}u} is the magnetic Laplace operator. Under a local assumption on the potential V , by combining variational methods, a penalization technique and the Ljusternik–Schnirelmann theory, we prove multiplicity properties of solutions and concentration phenomena for ϵ small. In this problem, the function f is only continuous, which allows to consider larger classes of nonlinearities in the reaction.

We study an anisotropic, possibly non-homogeneous version of the evolution 𝑝-Laplacian equation when fast diffusion holds in all directions. We develop the basic theory and prove symmetrization results from which we derive sharp L1L^{1}-L∞L^{\infty} estimates. We prove the existence of a self-similar fundamental solution of this equation in the appropriate exponent range, and uniqueness in a smaller range. We also obtain the asymptotic behaviour of finite mass solutions in terms of the self-similar solution. Positivity, decay rates as well as other properties of the solutions are derived. The combination of self-similarity and anisotropy is not common in the related literature. It is however essential in our analysis and creates mathematical difficulties that are solved for fast diffusions.

We investigate the multiplicity of subharmonic solutions for indefinite planar Hamiltonian systems J⁢z′=∇⁡H⁢(t,z){Jz^{\prime}=\nabla H(t,z)} from a rotation number viewpoint. The class considered is such that the behaviour of its solutions near zero and infinity can be compared two suitable positively homogeneous systems. Our approach can be used to deal with the problems in absence of the sign assumption on ∂⁡H∂⁡x⁢(t,x,y){\frac{\partial H}{\partial x}(t,x,y)}, uniqueness and global continuability for the solutions of the associated Cauchy problems. These systems may also be resonant. By the use of an approach of rotation number, the phase-plane analysis of the spiral properties of large solutions and a recent version of Poincaré–Birkhoff theorem for Hamiltonian systems, we are able to extend previous multiplicity results of subharmonic solutions for asymptotically semilinear systems to indefinite planar Hamiltonian systems.

In this paper, we consider the following Schrödinger–Poisson system with perturbation: { - Δ ⁢ u + u + λ ⁢ ϕ ⁢ ( x ) ⁢ u = | u | p - 2 ⁢ u + g ⁢ ( x ) , x ∈ ℝ 3 , - Δ ⁢ ϕ = u 2 , x ∈ ℝ 3 , \left\{\begin{aligned} \displaystyle-\Delta u+u+\lambda\phi(x)u&\displaystyle=% |u|^{p-2}u+g(x),&&\displaystyle x\in\mathbb{R}^{3},\\ \displaystyle-\Delta\phi&\displaystyle=u^{2},&&\displaystyle x\in\mathbb{R}^{3% },\end{aligned}\right. where λ>0{\lambda>0}, p∈(3,6){p\in(3,6)} and the radial general perturbation term g⁢(x)∈Lpp-1⁢(ℝ3){g(x)\in L^{\frac{p}{p-1}}(\mathbb{R}^{3})}. By establishing a new abstract perturbation theorem based on the Bolle’s method, we prove the existence of infinitely many radial solutions of the above system. Moreover, we give the asymptotic behaviors of these solutions as λ→0{\lambda\to 0}. Our results partially solve the open problem addressed in [Y. Jiang, Z. Wang and H.-S. Zhou, Multiple solutions for a nonhomogeneous Schrödinger–Maxwell system in ℝ3\mathbb{R}^{3}, Nonlinear Anal. 83 2013, 50–57] on the existence of infinitely many solutions of the Schrödinger–Poisson system for p∈(2,4]{p\in(2,4]} and a general perturbation term g .

We investigate the ground states of 3-component Bose–Einstein condensates with harmonic-like trapping potentials in ℝ2{\mathbb{R}^{2}}, where the intra-component interactions μi{\mu_{i}} and the inter-component interactions βi⁢j=βj⁢i{\beta_{ij}=\beta_{ji}} (i,j=1,2,3{i,j=1,2,3}, i≠j{i\neq j}) are all attractive. We display the regions of μi{\mu_{i}} and βi⁢j{\beta_{ij}} for the existence and nonexistence of the ground states, and give an elaborate analysis for the asymptotic behavior of the ground states as βi⁢j↗βi⁢j*:=a∗+12⁢(a∗-μi)⁢(a∗-μj){\beta_{ij}\nearrow\beta_{ij}^{*}:=a^{\ast}+\frac{1}{2}\sqrt{{(a^{\ast}-\mu_{i% })(a^{\ast}-\mu_{j})}}}, where 0<μi<a∗:=∥w∥22{0<\mu_{i}<a^{\ast}:=\|w\|_{2}^{2}} are fixed and w is the unique positive solution of Δ⁢w-w+w3=0{\Delta w-w+w^{3}=0} in H1⁢(ℝ2){H^{1}(\mathbb{R}^{2})}. The energy estimation as well as the mass concentration phenomena are studied, and when two of the intra-component interactions are equal, the nondegeneracy and the uniqueness of the ground states are proved.

In this paper, we study the nonlinear Klein–Gordon systems arising from relativistic physics and quantum field theories { u t ⁢ t - u x ⁢ x + b ⁢ u + ε ⁢ v + f ⁢ ( t , x , u ) = 0 , v t ⁢ t - v x ⁢ x + b ⁢ v + ε ⁢ u + g ⁢ ( t , x , v ) = 0 , \left\{\begin{aligned} \displaystyle{}u_{tt}-u_{xx}+bu+\varepsilon v+f(t,x,u)&\displaystyle=0,\\ \displaystyle v_{tt}-v_{xx}+bv+\varepsilon u+g(t,x,v)&\displaystyle=0,\end{aligned}\right. where u,vu,v satisfy the Dirichlet boundary conditions on spatial interval [0,π][0,\pi], b>0b>0 and f,gf,g are 2⁢π2\pi-periodic in 𝑡. We are concerned with the existence, regularity and asymptotic behavior of time-periodic solutions to the linearly coupled problem as 𝜀 goes to 0. Firstly, under some superlinear growth and monotonicity assumptions on 𝑓 and 𝑔, we obtain the solutions (uε,vε)(u_{\varepsilon},v_{\varepsilon}) with time period 2⁢π2\pi for the problem as the linear coupling constant 𝜀 is sufficiently small, by constructing critical points of an indefinite functional via variational methods. Secondly, we give a precise characterization for the asymptotic behavior of these solutions, and show that, as ε→0\varepsilon\to 0, (uε,vε)(u_{\varepsilon},v_{\varepsilon}) converge to the solutions of the wave equations without the coupling terms. Finally, by careful analysis which is quite different from the elliptic regularity theory, we obtain some interesting results concerning the higher regularity of the periodic solutions.

In this paper, we study the following Schrödinger–Poisson equations: { - ε 2 ⁢ Δ ⁢ u + V ⁢ ( x ) ⁢ u + K ⁢ ( x ) ⁢ ϕ ⁢ u = | u | p - 2 ⁢ u , x ∈ ℝ 3 , - ε 2 ⁢ Δ ⁢ ϕ = K ⁢ ( x ) ⁢ u 2 , x ∈ ℝ 3 , \left\{\begin{aligned} &\displaystyle{-}\varepsilon^{2}\Delta u+V(x)u+K(x)\phi u% =\lvert u\rvert^{p-2}u,&\hskip 10.0ptx&\displaystyle\in\mathbb{R}^{3},\\ &\displaystyle{-}\varepsilon^{2}\Delta\phi=K(x)u^{2},&\hskip 10.0ptx&% \displaystyle\in\mathbb{R}^{3},\end{aligned}\right. where p∈(4,6){p\in(4,6)}, ε>0{\varepsilon>0} is a parameter, and V and K are nonnegative potential functions which satisfy the critical frequency conditions in the sense that infℝ3⁡V=infℝ3⁡K=0{\inf_{\mathbb{R}^{3}}V=\inf_{\mathbb{R}^{3}}K=0}. By using a penalization method, we show the existence of multi-bump solutions for the above problem, with several local maximum points whose corresponding values are of different scales with respect to ε→0{\varepsilon\rightarrow 0}. Moreover, under suitable local assumptions on V and K , we prove the uniqueness of multi-bump solutions concentrating around zero points of V and K via the local Pohozaev identity.

We prove some results about existence of connecting and closed geodesics in a manifold endowed with a Kropina metric. These have applications to both null geodesics of spacetimes endowed with a null Killing vector field and Zermelo’s navigation problem with critical wind.

This paper deals with maximum principles depending on the domain and ABP estimates associated to the following Lane–Emden system involving fractional Laplace operators: { ( - Δ ) s ⁢ u = λ ⁢ ρ ⁢ ( x ) ⁢ | v | α - 1 ⁢ v in  ⁢ Ω , ( - Δ ) t ⁢ v = μ ⁢ τ ⁢ ( x ) ⁢ | u | β - 1 ⁢ u in  ⁢ Ω , u = v = 0 in  ⁢ ℝ n ∖ Ω , \left\{\begin{aligned} \displaystyle(-\Delta)^{s}u&\displaystyle=\lambda\rho(x% )\lvert v\rvert^{\alpha-1}v&&\displaystyle\phantom{}\text{in }\Omega,\\ \displaystyle(-\Delta)^{t}v&\displaystyle=\mu\tau(x)\lvert u\rvert^{\beta-1}u&% &\displaystyle\phantom{}\text{in }\Omega,\\ \displaystyle u&\displaystyle=v=0&&\displaystyle\phantom{}\text{in }\mathbb{R}% ^{n}\setminus\Omega,\end{aligned}\right. where s,t∈(0,1){s,t\in(0,1)}, α,β>0{\alpha,\beta>0} satisfy α⁢β=1{\alpha\beta=1}, Ω is a smooth bounded domain in ℝn{\mathbb{R}^{n}}, n≥1{n\geq 1}, and ρ and τ are continuous functions on Ω¯{\overline{\Omega}} and positive in Ω. We establish some maximum principles depending on Ω. In particular, we explicitly characterize the measure of Ω for which the maximum principles corresponding to this problem hold in Ω. For this, we derived an explicit lower estimate of principal eigenvalues in terms of the measure of Ω. Aleksandrov–Bakelman–Pucci (ABP) type estimates for the above systems are also proved. We also show the existence of a viscosity solution for a nonlinear perturbation of the nonhomogeneous counterpart of the above problem with polynomial and exponential growths. As an application of the maximum principles, we measure explicitly how small |Ω|{\lvert\Omega\rvert} has to be to ensure the positivity of the obtained solutions.

We consider ground states of the nonlinear fractional Schrödinger equation with potentials ( - Δ ) s ⁢ u + V ⁢ ( x ) ⁢ u = f ⁢ ( x , u ) , s ∈ ( 0 , 1 ) , (-\Delta)^{s}u+V(x)u=f(x,u),\quad s\in(0,1), on the whole space ℝN{\mathbb{R}^{N}}, where V is a periodic non-negative nontrivial function on ℝN{\mathbb{R}^{N}} and the nonlinear term f has some proper growth on u . Under uniform bounded assumptions about V , we can show the existence of a ground state. We extend the result of Li, Wang, and Zeng to the fractional case.