Multiplicity of weak solutions for a class of nonuniformly elliptic equations of p -Laplacian type

This paper deals with the multiplicity of weak solutions in W 0 1 ( Ω ) to a class of nonuniformly elliptic equations of the form − div ( a ( x , ∇ u ) ) = h ( x ) | u | r − 1 u + g ( x ) | u | s − 1 u in a bounded domain Ω of R N . Here a satisfies | a ( x , ξ ) | ≦ c 0 ( h 0 ( x ) + h 1 ( x ) | ξ | p − 1 ) for all ξ ∈ R N , a.e. x ∈ Ω , h 0 ∈ L p p − 1 ( Ω ) , h 1 ∈ L loc 1 ( Ω ) , h 1 ( x ) ≧ 1 for a.e. x in Ω , 1 < r < p − 1 < s < ( N p − N + p ) / ( N − p ) .