Abstract. We obtain solvability conditions for some elliptic equations involving non Fredholm operators with the methods of spectral theory and scattering theory for Schrödinger type operators. One of the main results of the work concerns solvability conditions for the equation −∆u + V (x)u −au = f where a ≥ 0. They are formulated in terms of orthogonality of the function f to the solutions of the homogeneous adjoint equation.

International audienceSome models in population dynamics with intra-specific competition lead to integro-differential equations where the integral term corresponds to nonlocal consumption of resources [8][9]. The principal difference of such equations in comparison with traditional reaction-diffusion equation is that homogeneous in space solutions can lose their stability resulting in emergence of spatial or spatio-temporal structures [4]. We study the existence and global bifurcations of such structures. In the case of unbounded domains, transition between stationary solutions can be observed resulting in propagation of generalized travelling waves (GTW). GTWs are introduced in [18] for reaction-diffusion systems as global in time propagating solutions. In this work their existence and properties are studied for the integro-differential equation. Similar to the reaction-diffusion equation in the monostable case, we prove the existence of generalized travelling waves for all values of the speed greater or equal to the minimal one. We illustrate these results by numerical simulations in one and two space dimensions and observe a variety of structures of GTWs

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