Study On Boundary Value Problem Of Sublinear Elliptic Equation Based On Nehari Manifold Method

In this paper, we use the method of Nehari manifold to study the 2-sublinear elliptic equation boundary value problem Where Ω is a bounded region with smooth boundary in RN, b(x) is a known fun-ction, and V(u) ∈C2(R,R) satisfies the following 2-sublinear conditionThe corresponding energy functional of question (1) is Suppose λ1 is the first eigenvalue of-△. If λ<λ1 and for each fixed u∈W01,2(Ω), fu/(t)=∫Ω b(x)V(tu)dx is an up convex function (fu"(t)≤0) with respect to t>0,and if there exist at least a t1> 0 such that the fibering map Φu (t) satisfies Φu(t1)≤0.Then, by demonstration, question (1) has an unique solution on the subset S+(A) of the Nehari manifold S(λ).We also use the Z2 index theory to consider the p-sublinear elliptic equation boundary value problem Where Ω(?(RN, (N> 1) is a bounded and smooth region, W(x, u) ∈ Cl (Ω×R,R), and W (x,u)=αW/αu denotes the partial derivative of W(x,u) with respect to u.We suppose W(x,u) satisfies(i) W(x,u)=W(x,-u), (?)x∈Ω, u∈R;(ii) there exist b> 0, such that(iii) there exist constants p>0,M> 0, such thatWe futher assume that for the eigenvectors ul,u2,…um of λ1,λ2…λm of-△p, When 1<p< 2, we haveThen, question (2) has at least m distinct pairs (u(x),-u(x)) of nontrivial classical solutions.