ORIGINAL ARTICLE
Schrödinger–Poisson systems with zero mass in the Sobolev limiting case
Giulio Romani,
Corresponding Author
Giulio Romani
Dipartimento di Scienza e Alta Tecnologia, Università degli Studi dell'Insubria, Varese, Italy
RISM-Riemann International School of Mathematics, Varese, Italy
Correspondence
Giulio Romani, Dipartimento di Scienza e Alta Tecnologia, Università degli Studi dell'Insubria, Varese, Italy.
Email: [email protected]
Search for more papers by this authorGiulio Romani,
Corresponding Author
Giulio Romani
Dipartimento di Scienza e Alta Tecnologia, Università degli Studi dell'Insubria, Varese, Italy
RISM-Riemann International School of Mathematics, Varese, Italy
Correspondence
Giulio Romani, Dipartimento di Scienza e Alta Tecnologia, Università degli Studi dell'Insubria, Varese, Italy.
Email: [email protected]
Search for more papers by this authorAbstract
We study the existence of positive solutions for a class of systems which strongly couple a quasilinear Schrödinger equation driven by a weighted -Laplace operator and without the mass term, and a higher-order fractional Poisson equation. Since the system is set in , the limiting case for the Sobolev embedding, we consider nonlinearities with exponential growth. Existence is proved relying on the study of a corresponding Choquard equation in which the Riesz kernel is a logarithm, hence sign-changing and unbounded from above and below. This is in turn solved by means of a variational approximating procedure for an auxiliary Choquard equation, where the logarithm is uniformly approximated by polynomial kernels. Our results are new even in the planar case .
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