Abstract
The existence of nontrivial solutions of Kirchhoff type problems is proved by using the local linking method, and the new results do not require classical compactness conditions.
Highlights
This paper is concerned with the existence of nontrivial solutions of the following nonlocal Kirchhoff type problem:
Similar nonlocal problems model several biological systems where u describes a process which depends on the average of itself, for example, that of the population density; see [1]
Some early classical studies of Kirchhoff equations were those of Bernstein [3] and Pohozaev [4]
Summary
This paper is concerned with the existence of nontrivial solutions of the following nonlocal Kirchhoff type problem:. Later Perera and Zhang [9, 10] obtained nontrivial solutions of Kirchhoff type problems with asymptotically 4-linear term via Yang index. In [11], Mao and Zhang used minimax methods and invariant sets of decent flow to prove 4-superlinear Kirchhoff type problems without the PS condition and got multiple solutions. Motivated by [12–15] and by the new information concerning local linking, we study Kirchhoff type problems 4superlinear growth as |u| → ∞. Most of the results on the existence and multiplicity of solutions of (1) were obtained under the above superlinear condition (AR) with or without the evenness assumption. Compared with the method of invariant sets of descent flow and Morse theory, Yang index, our method is more simple and direct
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