### A complement to the Fredholm theory of elliptic systems on bounded domains.

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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.

The existence of a positive solution for the generalized predator-prey model for two species $$\begin{array}{c}\Delta u+u(a+g(u,v\left)\right)=0\phantom{\rule{1.0em}{0ex}}\text{in}\phantom{\rule{4pt}{0ex}}\Omega ,\\ \Delta v+v(d+h(u,v\left)\right)=0\phantom{\rule{1.0em}{0ex}}\text{in}\phantom{\rule{4pt}{0ex}}\Omega ,\\ u=v=0\phantom{\rule{1.0em}{0ex}}\text{on}\phantom{\rule{4pt}{0ex}}\partial \Omega ,\end{array}$$ 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.

For a class of asymptotically periodic Schrödinger-Poisson systems with critical growth, the existence of ground states is established. The proof is based on the method of Nehari manifold and concentration compactness principle.

This paper is concerned with the following periodic Hamiltonian elliptic system $\{\phantom{\rule{3.33333pt}{0ex}}-\Delta \varphi +V\left(x\right)\varphi ={G}_{\psi}(x,\varphi ,\psi )\phantom{\rule{4pt}{0ex}}\text{in}\phantom{\rule{4.0pt}{0ex}}{\mathbb{R}}^{N},-\Delta \psi +V\left(x\right)\psi ={G}_{\varphi}(x,\varphi ,\psi )\phantom{\rule{4pt}{0ex}}\text{in}\phantom{\rule{4.0pt}{0ex}}{\mathbb{R}}^{N},\varphi \left(x\right)\to 0\phantom{\rule{4pt}{0ex}}\text{and}\phantom{\rule{4.0pt}{0ex}}\psi \left(x\right)\to 0\phantom{\rule{4pt}{0ex}}\text{as}\phantom{\rule{4.0pt}{0ex}}|x|\to \infty .$ Assuming the potential V is periodic and 0 lies in a gap of $\sigma (-\Delta +V)$, $G(x,\eta )$ is periodic in x and asymptotically quadratic in $\eta =(\varphi ,\psi )$, existence and multiplicity of solutions are obtained via variational approach.