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Abstract. In the present paper, by applying variant mountain pass theorem and Ekeland variational principle we study the existence of multiple nontrivial solutions for a class of Kirchhoff type problems with concave nonlinearityA new existence theorem and an interesting corollary of four nontrivial solutions are obtained. Mathematics Subject Classification (2000). 35J60, 58E30.

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In this paper, we study the following fourth-order elliptic equation with Kirchhoff-typewhere the constants a > 0, b ≥ 0. By constraint variational method and quantitative deformation lemma, we obtain that the problem possesses one least energy sign-changing solution u b . Moreover, we also prove that the energy of u b is strictly larger than two times the ground state energy. Finally, we give a convergence property of u b when b as a parameter and b → 0.2000 Mathematics Subject Classification. 35J50, 35J60.

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This paper is concerned with the following nonlocal elliptic system of (p,q)-Kirchhoff type−[M1(∫Ω|∇u|p)]p−1Δpu=λFu(x,u,v), in Ω,−[M2(∫Ω|∇v|q)]q−1Δqv=λFv(x,u,v), in Ω,u=v=0, on∂Ω.Under bounded condition onMand some novel and periodic condition onF, some new results of the existence of two solutions and three solutions of the above mentioned nonlocal elliptic system are obtained by means of Bonanno's multiple critical points theorems without the Palais-Smale condition and Ricceri's three critical points theorem, respectively.