Abstract
In this paper, we obtain the existence of positive solution for a system of quasilinear Schrödinger equations with concave nonlinearities which is related to several applications in Hydrodynamics, Heidelberg Ferromagnetism and Magnus Theory, Condensed Matter Theory, Dissipative Quantum Mechanics and nanotubes and fullerene-related structures. The quasilinear Schrödinger problem is studied by considering a suitable change of variables which transforms the original problem in to a semilinear one. By means of the several properties of the change of variables, constructions of suitable sub-supersolutions, monotonic iteration arguments and the Galerkin method, we obtain the existence of solution for the semilinear problem. The paper is divided in two parts. In the first one, we use the method of sub-supersolutions to obtain a solution for the problem. In the second part, we use the Galerkin method and a comparison argument to obtain a solution for the system considered. An important feature is that the sub-supersolution approach is rare in the literature for the type of problem considered here and the Galerkin method was not used to consider quasilinear Schrödinger equations.
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Corrêa, F.J.S.A., dos Santos, G.C.G. & Tavares, L.S. Solution for nonvariational quasilinear elliptic systems via sub-supersolution technique and Galerkin method. Z. Angew. Math. Phys. 72, 99 (2021). https://doi.org/10.1007/s00033-021-01532-8
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DOI: https://doi.org/10.1007/s00033-021-01532-8