The paper considers the existence of multiple solutions of the singular nonlocal elliptic problem −M(∫Ω | x|−ap | ∇u|p)div(|x|−ap | ∇u|p−2∇u) = λh(x) | u|r−2 u, x ∈ Ω, M(∫Ω | x|−ap | ∇u|p) | x|−ap | ∇u|p−2 (∂u/∂ν) = g(x) | u|q−2u, on ∂Ω, where 1 < (N + 1)/2 < p < N, a < (N − p)/p. By the variational method on the Nehari manifold, we prove that the problem has at least two positive solutions when some conditions are satisfied.
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机译:本文考虑奇异非局部椭圆问题−M(∫Ω | x | -ap sup> |∇u| p sup>)div(| x | < sup> -ap sup> |∇u| p − 2 sup>∇u)=λh(x)| u | r − 2 sup> u,x∈Ω, M em>(∫Ω | x em> | - ap em> > sup> |∇ u em> | p em> sup>)| x em> | − ap em> sup> | ∇ u em> | p em> −2 sup>(∂ u em> /∂ν) em> = < em> g em>( x em>)| u em> | q em> −2 sup> u em>,在Ω上,其中1 <( N em> > + 1)/ 2 em> p em> em> N em>, a em> <( N em>- p em>)/ p em>。通过Nehari流形上的变分方法,我们证明了当满足某些条件时,该问题至少具有两个正解。
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