Asymptotically Linear Elliptic Systems

Deterministic Chaos Theory: Basic Concepts

“…This equation is a miracle of brevity, relating a fluid's velocity, pressure, density and viscosity [20]. Since Eq.…”

“…The term ordinary is by hand or by computer, may give approximate solutions of ODE. One extremely popular is the Runge-Kutta method [20]. NLODE can exhibit very complicated behavior over extended time intervals, characteristic of chaos.…”

“…NLODE can exhibit very complicated behavior over extended time intervals, characteristic of chaos. The questions of existence and uniqueness of solutions of NLODE and PDE are hard problems and their resolution are of fundamental importance to the mathematical theory [20]. However, if the differential equation is a correctly formulated representation of a meaningful physical process, then one expects it to have a unique solution [18].…”

Infinitely many solutions for asymptotically linear periodic Hamiltonian elliptic systems

“…Their results were generalized by Sirakov [25] in a different way. Later, Bartsch and De Figueiredo [6] proved that the system admits infinitely many radial as well as non-radial solutions if G is even in z. Li and Yang [21] proved, via a generalized linking theorem, that (ES ) has a positive ground state solution for V ≡ 1 and an asymptotically quadratic nonlinearity G(x, ϕ, ψ) = F (ϕ)+ H(ψ), and based on this result they obtained a positive solution for G(x, ϕ, ψ) = ϕ 0 f (x, t)dt + ψ 0 (x, s)ds if f (x, ϕ) and g(x, ψ) have autonomous limitsf (ϕ) andh(ψ) at infinity. Very recently, Ding and Lin [16] considered semiclassical problems for systems of Schrödinger equations with subcritical and critical nonlinearities.…”

“…Clearly, {z j } is a (C) θ -sequence of Φ, hence is bounded by Lemma 4. [21] or Proposition 3.8 in Li and Szulkin [20] for more details. Hencẽ z is also a least energy solution.…”