New asymptotically quadratic conditions for Hamiltonian elliptic systems

This paper is concerned with the following Hamiltonian elliptic system − Δ u + V ( x ) u = W v ( x , u , v ) , x ∈ R N , − Δ v + V ( x ) v = W u ( x , u , v ) , x ∈ R N , $$ \left\{ \begin{array}{ll} -\Delta u+V(x)u=W_{v}(x, u, v), \ \ \ \ x\in {\mathbb{R}}^{N},\\ -\Delta v+V(x)v=W_{u}(x, u, v), \ \ \ \ x\in {\mathbb{R}}^{N},\\ \end{array} \right. $$ where z = (u, v) : ℝ N → ℝ2, V(x) and W(x, z) are 1-periodic in x. By making use of variational approach for strongly indefinite problems, we obtain a new existence result of nontrivial solution under new conditions that the nonlinearity W ( x , z ) := 1 2 V ∞ ( x ) | A z | 2 + F ( x , z ) $ W(x,z):=\frac{1}{2}V_{\infty}(x)|Az|^2+F(x, z) $ is general asymptotically quadratic, where V ∞(x) ∈ (ℝ N , ℝ) is 1-periodic in x and infℝ N V ∞(x) > minℝ N V(x), and A is a symmetric non-negative definite matrix.

Ground states and multiple solutions for Hamiltonian elliptic system with gradient term

This paper is concerned with the following nonlinear Hamiltonian elliptic system with gradient term $$\begin{array}{} \displaystyle \left\{\,\, \begin{array}{ll} -{\it\Delta} u +\vec{b}(x)\cdot \nabla u+V(x)u = H_{v}(x,u,v)\,\,\hbox{in}\,\mathbb{R}^{N},\\[-0.3em] -{\it\Delta} v -\vec{b}(x)\cdot \nabla v +V(x)v = H_{u}(x,u,v)\,\,\hbox{in}\,\mathbb{R}^{N}.\\ \end{array} \right. \end{array}$$ Compared with some existing issues, the most interesting feature of this paper is that we assume that the nonlinearity satisfies a local super-quadratic condition, which is weaker than the usual global super-quadratic condition. This case allows the nonlinearity to be super-quadratic on some domains and asymptotically quadratic on other domains. Furthermore, by using variational method, we obtain new existence results of ground state solutions and infinitely many geometrically distinct solutions under local super-quadratic condition. Since we are without more global information on the nonlinearity, in the proofs we apply a perturbation approach and some special techniques.

Concentration behavior of semiclassical solutions for Hamiltonian elliptic system

In this paper, we study the following nonlinear Hamiltonian elliptic system with gradient term $$\begin{array}{} \displaystyle \left\{ \begin{array}{ll} -\epsilon^{2}{\it\Delta} \psi +\epsilon \vec{b}\cdot \nabla \psi +\psi+V(x)\varphi=f(|\eta|)\varphi~~\hbox{in}~\mathbb{R}^{N},\\ -\epsilon^{2}{\it\Delta} \varphi -\epsilon \vec{b}\cdot \nabla \varphi +\varphi+V(x)\psi=f(|\eta|)\psi~~\hbox{in}~\mathbb{R}^{N},\\ \end{array} \right. \end{array}$$ where η = (ψ, φ) : ℝN → ℝ2, ϵ is a small positive parameter and b⃗ is a constant vector. We require that the potential V only satisfies certain local condition. Combining this with other suitable assumptions on f, we construct a family of semiclassical solutions. Moreover, the concentration phenomena around local minimum of V, convergence and exponential decay of semiclassical solutions are also explored. In the proofs we apply penalization method, linking argument and some analytical techniques since the local property of the potential and the strongly indefinite character of the energy functional.

On the Ground State to Hamiltonian Elliptic System with Choquard’s Nonlinear Term

In the present paper, we consider the following Hamiltonian elliptic system with Choquard’s nonlinear term −Δu+Vxu=∫ΩGvy/x−yβdygv in Ω,−Δv+Vxv=∫ΩFuy/x−yαdyfu in Ω,u=0,v=0 on ∂Ω,where Ω⊂ℝN is a bounded domain with a smooth boundary, 0<α<N, 0<β<N, and F is the primitive of f, similarly for G. By establishing a strongly indefinite variational setting, we prove that the above problem has a ground state solution.

Ground state solutions for Hamiltonian elliptic systems with super or asymptotically quadratic nonlinearity

This article concerns the Hamiltonian elliptic system: $$ \textstyle\begin{cases} -\Delta \varphi +V(x)\varphi =G_{\psi }(x,\varphi ,\psi ) & \mbox{in } \mathbb {R}^{N}, \\ -\Delta \psi +V(x)\psi =G_{\varphi }(x,\varphi ,\psi ) & \mbox{in } \mathbb {R}^{N}, \\ \varphi , \psi \in H^{1}(\mathbb {R}^{N}). \end{cases} $$ { − Δ φ + V ( x ) φ = G ψ ( x , φ , ψ ) in R N , − Δ ψ + V ( x ) ψ = G φ ( x , φ , ψ ) in R N , φ , ψ ∈ H 1 ( R N ) . Assuming that the potential V is periodic and 0 lies in a spectral gap of $\sigma (-\Delta +V)$ σ ( − Δ + V ) , least energy solution of the system is obtained for the super-quadratic case with a new technical condition, and the existence of ground state solutions of Nehari–Pankov type is established for the asymptotically quadratic case. The results obtained in the paper generalize and improve related ones in the literature.

Ground state solutions of Hamiltonian elliptic systems in dimension two

The aim of this paper is to study Hamiltonian elliptic system of the form 0.1$$\left\{ {\matrix{ {-\Delta u = g(v)} & {{\rm in}\;\Omega,} \cr {-\Delta v = f(u)} & {{\rm in}\;\Omega,} \cr {u = 0,v = 0} & {{\rm on}\;\partial \Omega,} \cr } } \right.$$ where Ω ⊂ ℝ2 is a bounded domain. In the second place, we present existence results for the following stationary Schrödinger systems defined in the whole plane 0.2$$\left\{ {\matrix{ {-\Delta u + u = g(v)\;\;\;{\rm in}\;{\open R}^2,} \cr {-\Delta v + v = f(u)\;\;\;{\rm in}\;{\open R}^2.} \cr } } \right.$$We assume that the nonlinearities f, g have critical growth in the sense of Trudinger–Moser. By using a suitable variational framework based on the generalized Nehari manifold method, we obtain the existence of ground state solutions of both systems (0.1) and (0.2).

Hamiltonian elliptic systems in dimension two with potentials which can vanish at infinity

We will focus on the existence of nontrivial solutions to the following Hamiltonian elliptic system [Formula: see text] where [Formula: see text] is a positive function which can vanish at infinity and be unbounded from above and [Formula: see text] and [Formula: see text] have exponential growth range. The proof involves a truncation argument combined with the linking theorem and a finite-dimensional approximation.