Existence And Multiplicity Of Solutions For A Class Of Elliptic Equations

In this paper, we consider the following elliptic equations with Dirichlet boundary value condition:andwhereΔ_pu is the p-Laplacian operator, that is,Δ_pu=div(|â–½u|^p-2â–½u) with p＞1 and equal toΔu when p=2, f∈C((?)×R, R),λ＞0 is a parameter. In our discussion, we always suppose p＞1 andΩis a bounded domain in R^N (N≥1) with smooth boundary (?)Ω.Ambrosetti and Rabinowitz have got the solutions of problem (1) by a very famous Mountain Pass Theorem in 1973, but they supposed the well-known AR condition hold, that is, for someμ＞p, M＞0,0＜μF(x, s)≤f(x, s)s, for all |s|≥M, x∈Ωwhere F(x, t)=∫₀^t f(x, s)ds. This (AR) condition usually plays a very important role in verifying that the corresponding functional has a Mountain-Pass geometry and showing a related (PS)_c sequence is bounded. In this paper, we will get the existence and multiplicity of solutions for problem (1) and (2) without AR condition. In order to get the solutions of problem (1) and (2), we change them into the critical points of the corresponding functionals. By the methods of the critical points, we can get the existence and multiplicity of the solutions of problem (1) and (2). Our main results are the following: Theorem 1 Suppose p=2 in problem (1) and f satisfies(f1) f is superlinear with respect to t at infinity, that is, (?)(f(x, t))/t=+∞a.e on x∈(?);(f2) f is subcritical in t, that is, there exists q∈(2, (2n)/(n-2)) when N＞2 and q∈(2, +∞) when N≤2 such that (?)(f(x, t))/(|t|^q-1=0 a.e on x∈(?);(f3) (?)(f(x, t))/t=α＜∞a.e on x∈(?), where 0≤α＜∞is a constant;(P) there existsθ≥1 such thatθG(x, t)≥G(x, st) for all x∈Ω, t∈R and s∈[0, 1], where G(x, t)=f(x, t)t-2F(x, t).Then there has at least one nontrivial solutions for problem(1).Theorem 2 Suppose f satisfies(H1) f(x, t)≥0 for all t≥0, x∈(?) and f(x, t)≡0 for all t≤0, x∈(?);(f1') (?)(f(x, t))/(t^p-1=+∞a.e on x∈(?);(f2') there exists q∈(p, (np)/(n-p)) when N＞p and q∈(p, +∞) when N≤p such that (?)(f(x, t))/(t^q-1=0 a.e on x∈(?);(f4) (?)(f(x, t)/(t^p-1)=α(x) and (?)(f(x, t))/(t^p-1)=+∞a.e on x∈(?), whereα∈L^∞((?)) satisfiesα(x)≤λ₁ for all x∈(?) andα(x)＜λ₁ on someΩ'(?)Ωwith positive measure andλ₁ is the first eigenvalue of -Δ_p;(P') there existsθ≥1 such thatθG(x, t)≥G(x, st) for all x∈Ω, t∈R and s∈[0, 1], where G(x, t)=f(x, t)t-pF(x, t).Then problem (1) has at least one positive solution.Theorem 3 Suppose f satisfies (f2'), (f4), (P') and(H2) f(x, t)t≥0 for all t≥0 and x∈(?);(f1*) (?)(f(x, t)t)/(|t|^p)=+∞a.e on x∈(?).Then problem (1) has at least two solutions, in which one is positive and the other is negative.Theorem 4 If h∈L^∞(Ω), setthen we haveΛ_h＞0 andφ_h∈H₀¹(Ω) whirφ_h＞0 a.e onΩsuch that∫_Ω|φ_h|²dx=Λ_h,∫_Ωh(x)|φ_h|²dx=1. Suppose f satisfies (H1) and (F) (?)(f(x, t))/t=b(x)(?)0且(?)(f(x, t))/t=d(x)(?)0 a.e on x∈Ω, where a, b∈L^∞(Ω).Then problem (2) has a positive solution whenΛ_d＜λ＜Λ_b orΛ_b＜λ＜Λ_d.Theorem 5 Suppose f satisfies (H2) and(F') (?)(f(x, t)/t=b(x)(?)0且(?)(f(x, t))/t=d(x)(?)0 a.e on x∈Ω, where b, d∈L^∞(Ω).Then problem (2) has two solutions ifΛ_d＜λ＜Λ_b orΛ_b＜λ＜Λ_d, in which one is positive and the other is negative.Theorem 6 Suppose f satisfies (H1), (f2) and(f5) (?)(f(x, t))/t=+∞uniformly in a.e x∈(?).Then there existsα＞0 such that problem (2) has at least one positive solution whenλ∈(0,α).Theorem 7 Suppose f satisfies (H1), (f1), (f2), (f5) and (P), then problem (2) has at least two positive solutions whenλ∈(0,α).