| Popis: |
In this paper we study the existence of nontrivial ground state solutions for the following class of p -Laplacian type equation − div a ( x , ∇ u ) + V ( x ) | x | − α p ∗ | u | p − 2 u = K ( x ) | x | − α p ∗ f ( u ) in R N , where 1 p N , N ≥ 3 , − ∞ α N − p p , α ≤ e ≤ α + 1 , d = 1 + α − e , p ∗ ≔ p ∗ ( α , e ) = N p N − d p (critical Hardy–Sobolev exponent); f has a quasicritical growth; V and K are nonnegative potentials; the function a satisfies | a ( x , ∇ u ) | ≤ c 0 | x | − α p h 0 ( x ) | ∇ u | p − 1 + c 0 ( 1 + | x | − α p ) h 1 ( x ) | ∇ u | p − 1 for any ξ ∈ R N , a.e. x ∈ R N , for any two positive functions h 1 ∈ L l o c ∞ ( R N ) , h 0 ∈ L α ¯ p p − 1 ( R N ) , with α ¯ = α p p ∗ . |
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