Multiplicity of Positive Solutions for a p - q -Laplacian Type Equation with Critical Nonlinearities

الملخص EN

We study the effect of the coefficient f ( x ) of the critical nonlinearity on the number of positive solutions for a p - q -Laplacian equation.

Under suitable assumptions for f ( x ) and g ( x ) , we should prove that for sufficiently small λ > 0 , there exist at least k positive solutions of the following p - q -Laplacian equation, - Δ p u - Δ q u = f x u | p * - 2 u + λ g x u | r - 2 u in Ω , u = 0 on ∂ Ω, where Ω ⊂ R N is a bounded smooth domain, N > p , 1 < q < N ( p - 1 ) / ( N - 1 ) < p ≤ max { p , p ^ * - q / ( p - 1 ) } < r < p ^ * , p ^ * = N p / ( N - p ) is the critical Sobolev exponent, and Δ s u = d i v ( | ∇ u | s - 2 ∇ u is the s -Laplacian of u .