Multiple Solutions For A Class Of Semilinear Elliptic Boundary Value Problems With Concave And Convex Nonlinearities

In this thesis,we will study the existence of infinitely many solutions of the following semilinear elliptic boundary value problem with cancave and convex nonlinearities: whereΩis a bounded domain of RN (N≥3)with the smooth boundary (?)Ω,λ>0 is a parameter,0<q<1,f(x,t),considered as a perturbation of the nonlinear term near the origin,satisfies: (A)f(x,t)∈C(Ω×R,R) and there is a p:1<p<∞such that (?)(f(x,t))/(|t|p)=0 uniformly in x∈Ω.We note that f(x,t) is not necessary to be odd in t (this is so called "perturbation from symmetry" problem),and,except the continuity, there is no growth condition on f for |t| large.By using an idea that developed by Bahri-Berestycki[4],P.H.Robino-witz[16] andM. Struwe[18],which was used to deal with superlinear "perturbation from symmetry" problems, we established an abstract theorem,with this theorem and the cut-off technique,we get the following result:Assume that N≥3,p,q satisfy max {0,(N(N-2)-4)/(N(N-2)+4)}<q<1 and 1+(N(1-q))/(1+q)< p<∞,f satisfies(A),then for eachλ>0,(Pλ) has infinitely many sign-changing solutions {uk} such that I(uk)<0 and I(uk)→0,‖uk‖→0,‖uk‖C(Ω)→0 as k→∞.