The Existence Of Solutions For Two Types Of Elliptic Equations With Concave-convex Nonlinear Terms

In this paper,we will discuss two nonlinear elliptic equations with different concave-convex nonlinear terms.First of all,considering the elliptic equation with concave-convex nonlinear terms:as for ??-?1,a more general elliptic equation(P2):will be taken into consideration.?(?)RN is a bounded domain with smooth boundary;?>0 is a parameter;?1 is the first eigenvalue of-? in H01(?);1<q<2 and f ? C(?× R,R).Due to lacking the(AR)condition and ??-?1 in(P2),we can't use the Mountain pass theorem to handle the problems in chapter two.We will first show that(P2)admits nontrivial solutions by using Local linking theorem under the(C)*condition.Then by applying the Fountain theorem under the(Cerami)condition to the elliptic equation(Pi)with concave-convex nonlinear terms,we prove the existence of infinitely many solutions.Secondly,we consider the nonlinear choquard equation with concave-convex nonlinear terms:where ?(?)RN is a bounded domain with smooth boundary;Ia is the Riesz potential;??(0,N);? is a parameter and ?>0.Since Choquard equation belongs to elliptic equation,we borrow the methods of elliptic equations which are used to deal with the concave-convex nonlinear terms.Under some different assumptions on f,g,we will prove the existence of infinitely many solutions and nontrivial solutions of(P3)under(PS)condition,when p?(N+?/N,N+?/(N-2)+)(N? 1)by using Mountain pass theorem,Fountain theorem and Hardy-Littlewood-Sobolev inequality in chapter three.