The Existence Of Positive Solutions For A Nonlinear Choquard Equation With Steep Potential

We consider the following Choquard function with steep potential(?)where N∈N,N≥3,α∈(0,N),p∈((N+α)/N,(N+α)/(N-2)),Vμ=1+μg(x),μ≥0,λ≥0.Kα(x)is the Riesz potential.The function f,g satisfy the following conditions(?)(?)When Vμ=1+μg(x),μ>0,λ=1,using the Ekeland variational principle and the Mountain Pass Lemma,we can obtain two positive solutions about the equation(M).When Vμ=1+μg(x),μ=0,λ>0,the equation(M)can get two positive solutions by using the critical point theory,the Ekeland variational principle,the Mountain Pass Lemma and the local Palais-Smale condition.When λ= 1,Vμ= 1 +μg(x)u>0,one hasTheorem 1.Assume thatN ≥ 3,α∈(0,N),p∈G((N+ α)/N,(N +α/(N-2)),and f,g satisfy(f1)-(f2),(g1)-(g3).There exist two constants μ,δ.Whenμ>μ*>0,|f|2<δ then(M)has at least two positive solutions.When λ>0,Vμ= 1 +μg(x)μ= 0,one getsTheorem 2.Assume thatN≥ 3,α∈(0,N),p ∈((N + α)/N,(N + α)/(N-2)),and f satisfies(f1)-(f2).Then there exists λ*>0 such that for all λ∈(0,λ*),(M)has at least two positive solutions.