Abstract
This work focuses on the symmetric solutions for a weighted quasilinear elliptic system involving multiple critical exponents in mathbb{R}^{N}. Based upon the Caffarelli-Kohn-Nirenberg inequality and the symmetric criticality principle due to Palais, we prove a variety of symmetric results under certain appropriate hypotheses on the singular potentials and the parameters.
Highlights
The present paper is dedicated to studying the following singular quasilinear elliptic system: ⎧ ⎪⎪⎨Lpβ u =K (x)|x|α p∗ (β,α )(μ ς |u|ς – u|v|τ +μ ς |u|ς – u|v|τ )h(x)|u|q– u in RN,⎪⎪⎩uL(xpβ)v, v=(xKp)∗(→ x(β)|x,α| α),(μ aτs ||ux||ς →|v|+τ ∞– v, + μ τ |u|ς |v|τ – v) + h(x)|v|q– v, in RN, ( . )where Lpβ – div(|x|β |∇ · |p– ∇·) is a quasilinear elliptic operator, < p < N, β ≤, N β
With respect to symmetric solutions for nonlinear elliptic systems, we remark that several symmetric results for singular problems were established in [ – ] and when
To obtain symmetric solutions for system (P K ), we prove the following local (PS)c condition, which is indispensable for the proof of Theorem
Summary
Since we are interested in problems with critical exponents and singular weighted functions, we refer the reader to With respect to symmetric solutions for nonlinear elliptic systems, we remark that several symmetric results for singular problems were established in [ – ] and when Α –β +p N +β –p for some > , where K+(∞) = lim sup|x|→∞ K+(x), problem (P K ) has at least one positive solution in (Dβ ,,pG(RN )) .
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