Existence And Multiplicity Of Solutions For Several Elliptic Systems

In the present paper,we study existence and multiplicity of solutions for several elliptic systems,by using variational methods,Lyapunov-Schmidt reduction method and Mountain-pass theory.Firstly,we deal with the existence of peak-solutions for a nonlocal elliptic system problem of FitzHugh-Nagumo type (?)where s ∈(0,1),N>2s,p ∈(1,N+2s/N-2s),Ω is a bounded domain in RN with Lipschitz boundary,and(-Δ)s is the fractional laplacian operator,ε is a small positive pa-rameter.By using the Lyapunov-Schmidt reduction method,we construct a single peak solution which is in the domain but near the boundary as ε sufficiently small.Then,we consider the following Kirchhoff type elliptic system (?)where Ω is a bounded domain in R2 containing the origin with smooth boundary,(?)∈[0.2),m is a Kirchhoff type function,‖ui‖2=∫Ω|▽ui|2dx,fi behaves like eβt2 as |t|→∞ for some β>0 and hi belongs to a suitable space,i=1,...,k,ε>0.By using the Ekeland variational principle and Mountain-pass theory with a suitable Trudinger-Moser inequality with singular weight,we obtain at least two solutions to this system as ε sufficiently small.Finally,we study systems as previous on Heisenberg group.Let HN=CN×R be the Heisenberg group,Q=2N+2 be the homogeneous dimension of HN,we consider the following Q-Laplacian subelliptic system (?)where Ω(?)HN is an open and bounded domain with smooth boundary and con-taining the origin,K is a Kirchhoff type function,0≤(?)<Q and λ is a positive parameter,and nonlinear terms Gu,Gv have critical exponential growth behave like exp(β|t|Q/Q-1)as |t|→∞ for some β>0.With some suitable assumptions,we obtain the existence of solutions for this system as λ sufficiently large.