Multiplicity Of Solutions For A Class Of General Elliptic Systems

In this paper, by using the mountain pass lemma and the "fountain theorem",we study two classes of general elliptic systems in divergence type.First,we consider the following elliptic systemswhereΩis a bounded smooth domain in R^N, N≥2. Under some suitable assumptionson a, b and F,we show the existence of multiple solutions of the problem. Our main results are as follows.(1) Suppose that function a, b and F satisfying conditions H₀, (F₁)- (F₄).Then the general elliptic systems have at least one pairs of nontrivial weak solution in W₀^1,p(Ω)×W₀^1,q{Ω).(2)Assume that a,b :Ω×R^N→R^N are odd with respect to the second variable, a(x,-s) =-a(x,s),b(x,-t) = -b(x,t), and functional F :Ω×R²â†’R is even with respect to the last variables, i.e. F(x, -s,-t) = F(x, s, t), and that conditions H₀, (F₁)-(F₃) and (F₄) are satisfied. Then the general elliptic systems have infinitely many pairs of solutions.Secondly, we are devoted to study the following general elliptic systemswhereΩis a bounded domain in R^N with smooth boundary and v is the outer normal vector on (?)Ω,λ(x),μ(x)∈L^∞(Ω) satisfying ess inf _x∈Ωλ(x),ess inf_x∈Ωμ(x)>0. We prove the existence of infinitely many pairs of solutions for the above problem by the "fountain theorem" and the " dual fountain theorem" respectively.