Multiplicity of Solutions for Gradient Systems Using Landesman-Lazer Conditions

and Applied Analysis 3 B∞ There is h ∈ C Ω,R such that ∣∣f x, z ∣∣ ≤ h x , ∀ x, z ∈ Ω × R2, ∣∣g x, z ∣∣ ≤ h x , ∀ x, z ∈ Ω × R2. 1.8 Under these hypotheses, system 1.1 is asymptotically quadratic at infinity due to the presence of a linear part given by the function A ∈ S2 Ω . In addition, when λk A 1 for some k ≥ 1 the problem 1.1 becomes resonant. In this case, in order to obtain existence and multiplicity of solutions for 1.1 , we will assume conditions of the Landesman-Lazer type introduced in the scalar case in 6 . These famous conditions are well known in the scalar case. However for gradient systems, to the best our knowledge, these conditions have not been explored in our case. In order to introduce our Landesman-Lazer conditions for system 1.1 , we need the following auxiliary assumptions. f∞ There are functions f , f −, f− , f−− ∈ C Ω such that f x lim u→∞ v→∞ f x, u, v , f − x lim u→∞ v→−∞ f x, u, v , f− x lim u→−∞ v→∞ f x, u, v , f−− x lim u→−∞ v→−∞ f x, u, v . 1.9 Moreover, g∞ there are functions g , g −, g− , g−− ∈ C Ω such that g x lim u→∞ v→∞ g x, u, v , g − x lim u→∞ v→−∞ g x, u, v , g− x lim u→−∞ v→∞ g x, u, v , g−− x lim u→−∞ v→−∞ g x, u, v , 1.10 where the limits in 1.9 and 1.10 are taken uniformly and for all x ∈ Ω. So we can write the Landesman-Lazer conditions for our problem 1.1 , when k 1. It will be assumed either