“…Alves [1], Ma-Rivera [14] and HeZou [11] studied the existence of positive solutions and infinitely many positive solutions of the problem (1.3) by variational methods, respectively; Perera and Zhang [17] obtained one nontrivial solutions of (1.3) by Yang index theory; Mao-Zhang [15], Zhang and Perera [20] got three nontrivial solutions (a positive solution, a negative solution, a sign changing solution) of (1.3) by invariant sets of descent flow; Cheng-Wu [6] obtained the existence results of positive solutions of problem (1.3), also in [7] they used a three critical point theorem due to Brezis-Nirenberg [4] and a Z 2 version of the Mountain Pass Theorem due to Rabinowitz [19] to study the existence of multiple nontrivial solutions of problem (1.3) under some weaker assumptions. In order to establish multiple solutions for problem (1.1), we make the following assumptions:…”
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.
“…Alves [1], Ma-Rivera [14] and HeZou [11] studied the existence of positive solutions and infinitely many positive solutions of the problem (1.3) by variational methods, respectively; Perera and Zhang [17] obtained one nontrivial solutions of (1.3) by Yang index theory; Mao-Zhang [15], Zhang and Perera [20] got three nontrivial solutions (a positive solution, a negative solution, a sign changing solution) of (1.3) by invariant sets of descent flow; Cheng-Wu [6] obtained the existence results of positive solutions of problem (1.3), also in [7] they used a three critical point theorem due to Brezis-Nirenberg [4] and a Z 2 version of the Mountain Pass Theorem due to Rabinowitz [19] to study the existence of multiple nontrivial solutions of problem (1.3) under some weaker assumptions. In order to establish multiple solutions for problem (1.1), we make the following assumptions:…”
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.
“…Mao and Zhang [14] obtained three solutions by the invariant sets of descent flow. He and Zou [7] showed existence of infinitely many solutions by using the local minimum methods and the fountain theorems. Cheng and Wu [5] studied the existence of positive solutions for problem (1.1) when the nonlinearity f is asymptotically t 3 -growth at infinity.…”
Abstract. In this paper, we consider the following Kirchhoff-type Schrödinger system, where a i and b i are positive constants for i = 1, 2, γ > 0 is a parameter, V (x) and W (x) are nonnegative continuous potential functions. By applying the Nehari manifold method and the concentration-compactness principle, we obtain the existence and concentration of ground state solutions when the parameter γ is sufficiently large.
“…As for in nitely many solutions, we refer readers to [9] [10]. He and Zou [11] considered the class of Kirchhoff type problem when (F) f is a continuous function satisfies: …”
In this paper, we establish the existence of at least four distinct solutions to an Kirchhoff type problems involving the critical Caffareli-Kohn-Niremberg exponent, concave term and sign-changing weights, by using the Nehari manifold and mountain pass theorem.
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