New existence and multiplicity of nontrivial solutions for nonlocal elliptic Kirchhoff type problems

In the present paper, we study the existence and multiplicity of nontrivial solutions for nonlocal elliptic Kirchhoff type problems { − ( a + b ∫ Ω ∣ ∇ u ∣ 2 d x ) Δ u = f ( x , u ) , in Ω ; u = 0 , on ∂ Ω , where a , b > 0 are real numbers. Some new existence and multiplicity results of nontrivial solutions are obtained by means of the mountain pass theorem and a cut-off technique. One known result is improved.