Abstract
In this paper, we consider singular elliptic systems involving a strongly coupled critical potential and concave nonlinearities. By using variational methods and analytical techniques, the existence and multiplicity of positive solutions to the system are established.MSC: 35J60, 35B33.
Highlights
Introduction and main resultsIn this paper, we consider the following elliptic system: ⎧ ⎪⎪⎨Lu = η α * |u|α – |v|β u + σ |u|q– u, ⎪⎪⎩Lv = u, η β * |u|α v ∈ H (
For any μ < μ, η + η >, ≤ ηi < ∞, αi, βi > and αi + βi = *, i =, by the Young and Sobolev inequalities, the following best constants are well defined on the space D = (D, (RN ) \ { }) : Sη,β (μ) inf (u,v)∈D
Throughout this paper, we always assume that the assumption (H) holds, u H := u μ =
Summary
Definitions of strongly and weakly coupled terms are as follows. The terms |u|α and |v|β (α, β > ) are weakly coupled, |Lu|α|Kv|β (α, β > ) is strongly coupled when L or K is a derivative operator. Η |u|α |v|β + η |u|α |v|β is strongly coupled when η and η are positive. J ∈ C (H × H, R) and the duality product between H × H and its dual space (H × H)– is defined as. A pair of functions (u, v) ∈ H × H is said to be a weak solution of L is positive and the first eigenvalue (μ) of L and the following best constant are well defined: S(μ) := inf u∈D , (RN )\{ }
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