Radial and nonradial solutions for nonautonomous Kirchhoff problems

Abstract

In this paper, we study the following nonautonomous Kirchhoff problem: where , , , is a positive constant, is a parameter, and the potential functions belong to . The existence of radially symmetric and positive solution to the above problem is first established for all when are radially symmetric and , and the range of can be extended to with the aid of a coercive type assumption on . Moreover, we show the existence of infinitely many solutions with high energies via the fountain theorem under more general assumption on which allows it to be sign‐changing. When and , we show that the above problem possesses infinitely many solutions with negative critical values for small provided that the function belongs to a suitable space. In particular, by imposing a hypothesis on the potential controlling its growth at infinity, we obtain a nonradial solution via the mountain pass theorem and the principle of symmetric criticality.