Multiple non semi‐trivial solutions of systems of Kirchhoff‐type equations with discontinuous nonlinearities in R N

In the present paper, we study the problem of multiple non semi‐trivial solutions for the following systems of Kirchhoff‐type equations with discontinuous nonlinearities 1.1− a + b ∫R N| ∇ u | 2 dx △ u + V ( x ) u = F u ( x , | u | , | v | ) u * , x ∈ R N ,− c + d ∫R N| ∇ v | 2 dx △ v + V ( x ) v = F v ( x , | u | , | v | ) v * , x ∈ R N ,u ( x ) → 0 and v ( x ) → 0 as | x | → ∞ ,where F ∈ C 1 ( R N × R + × R + , R ), V ∈ C ( R N , R ),u * ( x ) =u ( x ) | u ( x ) | , u ( x ) ≠ 0 ,| u * ( x ) | ≤ 1 , u ( x ) = 0 ,andv * ( x ) =v ( x ) | v ( x ) | , v ( x ) ≠ 0 ,| v * ( x ) | ≤ 1 , v ( x ) = 0 .By establishing a new index theory, we obtain some multiple critical point theorems on product spaces, and as applications, three multiplicity results of non semi‐trivial solutions for (1.1) are obtained. Copyright © 2015 John Wiley & Sons, Ltd.