Existence And Multiplicity To Solutions For A Fourth-order Elliptic Equation

This paper has been divided into three chapters:Chapter1is the preface.In Chapter2,we discuss fourth-order elliptic equation Navier boundary problem: where (?)(?) RN is a smooth open bounded domain,â–³2is the biharmonic operator, a <λ1is a constant, λ1is the first eigenvalue for(一△,H01((?))),and f∈((?)×R, R).We obtain the following results:Theorem2.1Suppose that f satisfies:(f1) f(x,t)≡0,x∈(?), t≤0, f(x,t)≥0,x∈(?), t>0;(f2) f(x,t)/t is nondecreasing with respect to t≥0, a.e. x∈(?);(f3) limtâ†'0+f(x,t)/t=p(x), limt(?)+â†'+∞f(x,t)/t=+∞uniformly for a.e. x∈(?), where p(x)≥0, x∈(?), p∈L∞((?)) and||p||∞<â–³1:=λ1(λ1-a);(f4) limtâ†'+∞f(x,t)/ts-1=0uniformly for a.e. x∈(?), where s>2if N=1,2,2<s <(2N-2)/(N-2) if N≥3. Then problem (2.1) has at least a positive solution.Theorem2.2Suppose that f(x,t)=u|t|q-2t+λ|t|p-2t, where1<q <2<p <2’. Then for all μ>0, λ∈R, problem (2.1)has a sequence of negative critical values which converge to0.Theorem2.3Suppose that f satisfies:(f5) There exists C0>0such that f(x,t)|≤C0(|t|p-1+1) for all (x,t)∈(?)×R, where(f6) For all (x,t)∈(?)×R\{0}, t f(x,t)>0, and limtâ†'0f(x,y)/t=+∞uniformly for a.e. x∈(?): (f7) f(x,-t)=-f(x,t) for all (x,t)∈(?)×R.Then problem (2.1) has a sequence of nontrivial solutions.In Chapter3,we discuss the following nonlinear elliptic equation:Suppose that f satisfies:(f8) f∈C(R,R);(fg) limtâ†'0f(t)/t=0;(f10) There exists l>0such that lim|t|â†'+∞f(t)/t=l;(f11) f(t)/t is strictly increasing on R\{0} in|t|. We obtain the following result:Theorem3.1suppose that(f8)-(f11) hold. If problem (3.1) has a mountain pass solution u0≠0, c1=I(u0), then c0=c1=cI.