Abstract
We consider the following nonlinear Neumann problem{−Δu−γu|x|2+μu=|u|2s⁎−2u|x|s in BR⊂RN,N≥3∂u∂ν=0 on ∂BR where γ<γ‾:=(N−2)24, 0<s<2, 2s⁎=2(N−s)N−2 and BR is the ball centered at the origin with radius R. Firstly, we establish the existence of infinitely many positive radial solutions which are singular at the origin. Secondly, we investigate the existence and regularity of a least-energy solution. Lastly, we study the symmetric properties of a regular least-energy solution.
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