Anmin Mao, Xiaorong Luo, Multiplicity of solutions to linearly coupled Hartree systems with critical exponent
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DOI: 10.23952/jnva.7.2023.2.01
Volume 7, Issue 2, 1 April 2023, Pages 173-200
Abstract. We consider the existence multiple solutions to the linearly
coupled elliptic system
in
in
in
on
where is a bounded domain with smooth boundary in
,
,
are constants,
is the first eigenvalue of
,
is a coupling parameter,
are nonnegative, and
is the upper critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality. We prove that the system has a positive ground state solution by mountain pass theorem for small
. By a perturbation argument, when
, comparing with the mountain pass type solution, another positive higher energy solution is obtained when
is small. In addition, the asymptotic behaviours of these solutions are analyzed as $\beta\rightarrow0$.
How to Cite this Article:
A. Mao, X. Luo, Multiplicity of solutions to linearly coupled Hartree systems with critical exponent, J. Nonlinear Var. Anal. 7 (2023), 173-200.