Multiple Positive Solutions for a Quasilinear Elliptic System Involving Concave-Convex Nonlinearities and Sign-Changing Weight Functions

Let Ω ∋ 0 be an-open bounded domain in â„ ð ‘ ( ð ‘ â‰¥ 3 ) and ð ‘ âˆ— = ( ð ‘ ð ‘ / ( ð ‘ âˆ’ ð ‘ ) ) . We consider the following quasilinear elliptic system of two equations in ð ‘Š 0 1 , ð ‘ ( Ω ) × ð ‘Š 0 1 , ð ‘ ( Ω ) : − Δ ð ‘ ð ‘¢ = 𠜆 ð ‘“ ( ð ‘¥ ) | ð ‘¢ | ð ‘ž − 2 ð ‘¢ + ( ð ›¼ / ( ð ›¼ + ð ›½ ) ) â„Ž ( ð ‘¥ ) | ð ‘¢ | ð ›¼ − 2 ð ‘¢ | ð ‘£ | ð ›½ , − Δ ð ‘ ð ‘£ = 𠜇 ð ‘” ( ð ‘¥ ) | ð ‘£ | ð ‘ž − 2 ð ‘£ + ( ð ›½ / ( ð ›¼ + ð ›½ ) ) â„Ž ( ð ‘¥ ) | ð ‘¢ | ð ›¼ | ð ‘£ | ð ›½ − 2 ð ‘£ , where 𠜆 , 𠜇 > 0 , Δ ð ‘ denotes the ð ‘ -Laplacian operator, 1 ≤ ð ‘ž < ð ‘ < ð ‘ , ð ›¼ , ð ›½ > 1 satisfy ð ‘ < ð ›¼ + ð ›½ ≤ ð ‘ âˆ— , and ð ‘“ , ð ‘” , â„Ž are continuous functions on Ω which are somewhere positive but which may change sign on Ω . We establish the existence and multiplicity results of positive solutions to (the above mentioned quasilinear elliptic system equations) by variational methods.