Multiple Solutions Of Neumann Problem Of Nonlinear Elliptic Equations With Stability

In this paper we study the existence of multiple solutions of a class of non-linear elliptic equation with Neumann boundary condition and the stability ofequilibrium solutions for the corresponding parabolic equation.Firstly we consider the existence of multiple solutions of the Neumannboundary-value problem.Let Ω Rⁿ be a bounded domain with smooth boundary Ω. Consider theNeumann boundary-value problemwhere α>0, f∈C¹(Ω,R) satisfies f(0)=0.Denote by σ(-Δ):={λi|0=λ₁<λ₂… ≤λ_k≤…} the eigenvalues of the following linear problemUnder the following assumptions:(i) There are constaifts C₁,C₂>0 such that(ii) There are 0>2, M>0 such that(iv) There are M>0, N>0 such that f(-N)=f(M)=0;(v) There is m>0 such that the function g(u)+mu is increasing.the Problem (I) has at least seven nonzero solutions in which two are posi-tive, two are negative and three are sign--changing.Secondly we consider the stability of eqllilibrium sollltions fOr the parabolicequation{t;::f;(>:j1::::;:<;:::) (II)which corresponds the Problem (I). It is well--known that solutions of (I) areequilibrium solutions of (II). When the function f satisfies the conditions (i) -(v)and the fOllowing(vi) u = M is the least positive valuc at which f is vanished and u = --Nis the largest negative value at which f is vanished.(vii) i(u) is strictly concave in (0, M] and is strictly convex in [--N, 0),we have that a pair of positive--negative sol1lti()ns of (I) are Iocally asyTnptoti(.allystable and the zero solution of (I) is unstable.The study of the problem (I) are based on the variational methods andcritical point theory. In view of the variational point, solutions of (I) are criticalpoints of corresponding functional defined ()11 the Hilbert space E f= 14/"'(fl),Let X t= {u C C'(O)l %i,. = 0} C E be a Banach space a11d [--N, M] (= {?, EXI -- N 5 u(x) 5 AI, x E fl} be the order interval in X. By using the sub-supsolution method, Mountain Pass Theorem in order intervals([1]), Leray-Schauderdegree theory and the invariance of decreasing flow, we obtain that the problem(I) haJs four nonzero solutions, in which one is positive, one is negative and twoare sign-changing, belonging to the order iIlteI'val [--N, Ml, aIld llas tllree IloIlzeI'()solutions outside of [--N, M], in which one is positive, one is negative and twoare sign-changing. By using the sub-sup solution methods and the linearizationmethods, we prove that a pair of positive-negative equilibrium solutions of (II)in [--N, M] are locally asymptotically stable. We can also prove that the trivialequilibrium solution of (II) is unstable.