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 λ≤-λ1,a more general elliptic equation(P2):will be taken into consideration.Ω(?)RN is a bounded domain with smooth boundary;μ>0 is a parameter;λ1 is the first eigenvalue of-Δ in H01(Ω);1<q<2 and f ∈ C(Ω× R,R).Due to lacking the(AR)condition and λ≤-λ1 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 Ω(?)RN is a bounded domain with smooth boundary;Ia is the Riesz potential;α∈(0,N);μ is a parameter and λ>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+α/N,N+α/(N-2)+)(N≥ 1)by using Mountain pass theorem,Fountain theorem and Hardy-Littlewood-Sobolev inequality in chapter three.