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A maximum principle for systems with variational structure and an application to standing waves

Nicholas D. Alikakos, Giorgio Fusco (2015)

Journal of the European Mathematical Society

We establish via variational methods the existence of a standing wave together with an estimate on the convergence to its asymptotic states for a bistable system of partial differential equations on a periodic domain. The main tool is a replacement lemma which has as a corollary a maximum principle for minimizers.

A predator-prey model with combined death and competition terms

Joon Hyuk Kang, Jungho Lee (2010)

Czechoslovak Mathematical Journal

The existence of a positive solution for the generalized predator-prey model for two species Δ u + u ( a + g ( u , v ) ) = 0 in Ω , Δ v + v ( d + h ( u , v ) ) = 0 in Ω , u = v = 0 on Ω , are investigated. The techniques used in the paper are the elliptic theory, upper-lower solutions, maximum principles and spectrum estimates. The arguments also rely on some detailed properties of the solution of logistic equations.

Infinitely many solutions for asymptotically linear periodic Hamiltonian elliptic systems

Fukun Zhao, Leiga Zhao, Yanheng Ding (2010)

ESAIM: Control, Optimisation and Calculus of Variations

This paper is concerned with the following periodic Hamiltonian elliptic system { - Δ ϕ + V ( x ) ϕ = G ψ ( x , ϕ , ψ ) in N , - Δ ψ + V ( x ) ψ = G ϕ ( x , ϕ , ψ ) in N , ϕ ( x ) 0 and ψ ( x ) 0 as | x | . Assuming the potential V is periodic and 0 lies in a gap of σ ( - Δ + V ) , G ( x , η ) is periodic in x and asymptotically quadratic in η = ( ϕ , ψ ) , existence and multiplicity of solutions are obtained via variational approach.


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