The Existence Of Multiple Solutions For A Class Of Semilinear Elliptic Equations, Dirichlet Problem

In this paper, we study the existence of multiple nontrivial solutions for a kind of elliptic resonant problems by applying minimax methods and Morse theory. Consider the boundary value problemwhere is a bounded open domain with smooth boundary , and f : is a C¹ function that satisfieswhere , the spectrum of with irichlet data. It follows from(1) and (2) that the problem (P) has a trivial solution u = 0 and is resonant nearinfinity.Corresponding to , the standard Sobolev space can be split whereWe assume the following conditions:such that then there exists and such thatwhere .Our main results in this paper are the following theorems.Theorem 1 Let f satisfy (f'), (1), (2) and f'{x,0) < λ₁, Then the problem (P) has at least three nontrivial solutions in each of the following cases:with (ii)(g^-) with k ≥ 3. In the next two theorems, we assume that and there exists such thatandTheorem 2 Let f satisfy (f'), (1), (2)and k = 1. Then problem (P) has at least two nontrivial solutions in each of the following cases:(i)(g⁺)(f₀⁺) with m≠1 ;(ii) (g⁺), (f₀^-) with m ≠ 2;(iii) (g^-), (f₀⁺) with m ≥ 1; (iv) (g^-), (f₀^-) with m ≠ 1.Theorem 3 Let f satisfy (f'), (1), (2) with k ≥ 2 and(f₁) there exists t₁≠ 0 such that f(x,t₁) = 0 for a.e. x ∈Ω. Then the problem (P) has at least four nontrivial solutions in each of the following cases:(i) (g⁺), (f₀⁺) with m≠k; (ii) (g⁺), (f₀^-) with m ≠ k + 1;(iii) (g^-), (f₀⁺) with m ≥ k -1; (iv) (g^-), (f₀^-) with m ≠k.