ON SOME SEMILINEAR ELLIPTIC PROBLEMS WITH SINGULAR POTENTIALS INVOLVING SYMMETRY

This paper deals with the existence and multiplicity of solutions for a class of semilinear elliptic problems of the form { − Δ u = u = μ | x | 2 u + f ( x , u ) 0 in on Ω , ∂ Ω , where Ω = Ω1× Ω2⊂ ℝ N (N≧ 5) is a bounded domain having cylindrical symmetry, Ω1⊂ ℝ m is a bounded regular domain and Ω2is ak-dimensional ball of radiusR, centered in the origin andm+k=N, andm≧ 2,k≧ 3, 0 ≦ μ < μ * = ( N − 2 2 ) 2 . The proofs rely essentially on the critical point theory tools combined with the Hardy inequality. 2000Mathematics Subject Classification: 35J65, 35J20. Key words and phrases: Semilinear elliptic problems, Singular potentials, Symmetry, Ekeland's variational principle, Mountain pass theorem.