SOLUTIONS FOR A NONHOMOGENEOUS ELLIPTIC PROBLEM INVOLVING CRITICAL SOBOLEV-HARDY EXPONENT IN RN

Abstract

For the following elliptic problem { - Δ u - μ u | x | 2 = | u | 2 * ( s ) - 2 u | x | s + h ( x ) , on R N u ∈ D 1 , 2 ( R N ) , N ≥ 3 , 0 ≤ μ < μ ¯ = ( N - 2 ) 2 4 , 0 ≤ s < 2 , where 2 * ( s ) = 2 ( N - s ) N - 2 is the critical Sobolev-Hardy exponent, h ( x ) ∈ ( D 1 , 2 ( R N ) * , the dual space of ( D 1 , 2 ( R N ) ) , with h( x)≥(≢)0. By Ekeland's variational principle, subsuper solutions and a Mountain Pass theorem, the authors prove that the above problem has at least two distinct solutions if ∥ h ∥ * < C N , s A s N - s 4 - 2 s ( 1 - μ μ ) 1 2 , C N , s = 4 - 2 s N - 2 ( N - 2 N + 2 - 2 s ) N + 2 - 2 s 4 - 2 s and A s = inf u ∈ D 1 , 2 ( R N ) \\ { 0 } ∫ R N ( | ∇ u | 2 - μ u 2 | x | 2 ) d x ( ∫ R N | u | 2 * ( s ) | x | s d x ) 2 2 * ( s ) .