Quasilinear asymptotically linear Schrödinger problem in R^N without monotonicity
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We establish existence and non-existence results for a quasilinear asymptotically linear Schrodinger problem. In the first result, we prove that a minimization problem constrained to the Pohozaev manifold is not achieved. In the second, the main argument consists in a splitting lemma for a functional constrained to the Pohozaev manifold. Because of the lack of the monotonicity we are not able to project to the usual Nehari manifold any longer, and this approach is crucial in order to compare the critical level to reach a contradiction. This argument was used in [21, 24, 32] for semilinear equations and in  for quasilinear equations.
CitationMiyagaki, O. H., Moreira, S. I., & Ruviaro, R. (2018). Quasilinear asymptotically linear Schrödinger problem in R^N without monotonicity. Electronic Journal of Differential Equations, 2018(164), pp. 1-21.
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