Existence And Multiplicity Of Solutions For A Class Of Elliptic Equations At Resonance

In this paper, we mainly consider the following elliptic equationshere p>1,â–³_pu = div(|â–½u|^p-2â–½u) ,Ωis a bounded smooth domain in R^N (N≥1), h satisfies some integrability condition,λ₁ is the first eigenvalue of -â–³_p in W₀^1,p(Ω) subject to Dirichlet zero boundary condition, g:(?)×R→R is a, Caratheodory function and satisfies(g₁)(?)(g(x,t)/(|t|^p-2t))=0 uniformly for x∈Ω(g₂) For every M > 0, there exists a function L_M∈L^p'(Ω) such that|g(x,t)|≤L_M(x)for all t∈R and a.e. x∈Ω.If (g₁) holds, problem (P) is called to be elliptic equation resonant at the first eigenvalue. In this paper, by considering a new Landesman-Lazer-type solvability condition, we discuss existence and multiplicity of solutions of problem (P) via critical point theory.Definewhere G(x, t)=integral from n=0 to t g(x, s)ds, 1≤q1(x,t) for short.The main results of this paper are following theorems:Theorem 1 Assume that h∈L^p' (Ω), g is a Caratheodory function and satisfies(g₁), (g₂), moreover, either (?)∈L^p' (Ω) such thator (?)∈L^p' (Ω)such thatThen problem (P) at least exist one solution in W.Theorem 2 Assume that h = 0, g is a Caratheodory function and satisfies (g₁),(g₂), moreover, either (?)∈L^∞(Ω) such thatfor some q∈[1, p). Then problem (P) at least exist one solution in W.In addition, we suppose that(g₃) (?)(g(x,s))/(|s|^p-2s)=a₀;(?)(g(x,s))/(|s|^p-2s)=d₀Theorem 3 Assume that h = 0, g is a Caratheodory function and satisfies (g₁), (g₂), (g₃) with a₀,d₀<0, additional, g(x,0) = 0, and (3) holds for some q∈[1, p). Then problem (P) at least exist two nontrivial solutions in W.