Strongly indefinite functionals and multiple solutions of elliptic systems

Abstract

We study existence and multiplicity of solutions of the elliptic system \[ { − Δ u = H u ( x , u , v ) a m p ; in  Ω , − Δ v = − H v ( x , u , v ) a m p ; in  Ω , u ( x ) = v ( x ) = 0 on  ∂ Ω , \begin {cases} -\Delta u =H_u(x,u,v) & \text {in $\Omega $}, \\ -\Delta v =-H_v(x,u,v) & \text {in $\Omega $}, \quad u(x) = v(x) = 0 \quad \text {on $\partial \Omega $}, \end {cases} \] where Ω ⊂ R N , N ≥ 3 \Omega \subset \mathbb {R}^N, N\geq 3 , is a smooth bounded domain and H ∈ C 1 ( Ω ¯ × R 2 , R ) H\in \mathcal {C}^1(\bar {\Omega }\times \mathbb {R}^2, \mathbb {R}) . We assume that the nonlinear term \[ H ( x , u , v ) ∼ | u | p + | v | q + R ( x , u , v )     with     lim | ( u , v ) | → ∞ R ( x , u , v ) | u | p + | v | q = 0 , H(x,u,v)\sim |u|^p + |v|^q + R(x,u,v) \ \ \text {with} \ \ \lim _{|(u,v)|\to \infty }\frac {R(x,u,v)}{|u|^p+|v|^q}=0, \] where p ∈ ( 1 ,   2 ∗ ) p\in (1, \ 2^*) , 2 ∗ := 2 N / ( N − 2 ) 2^*:=2N/(N-2) , and q ∈ ( 1 ,   ∞ ) q\in (1, \ \infty ) . So some supercritical systems are included. Nontrivial solutions are obtained. When H ( x , u , v ) H(x,u,v) is even in ( u , v ) (u,v) , we show that the system possesses a sequence of solutions associated with a sequence of positive energies (resp. negative energies) going toward infinity (resp. zero) if p > 2 p>2 (resp. p > 2 p>2 ). All results are proved using variational methods. Some new critical point theorems for strongly indefinite functionals are proved.