Radial and nonradial solutions for nonautonomous Kirchhoff problems.

In this paper, we study the following nonautonomous Kirchhoff problem: −1+b∫ℝN|∇u|2dxΔu+V(x)u=a(x)|u|p−2u+λ|u|q−2u,x∈ℝN,u∈H1ℝN,$$ \left\{\begin{array}{l}-\left(1+b{\int}_{{\mathrm{\mathbb{R}}}^N}{\left|\nabla u\right|}^2\mathrm{d}x\right)\Delta u+V(x)u=a(x){\left|u\right|}^{p-2}u+\lambda {\left|u\right|}^{q-2}u,x\in {\mathrm{\mathbb{R}}}^N,\\ {}u\in {H}^1\left({\mathrm{\mathbb{R}}}^N\right),\end{array}\right. $$where N≥2$$ N\ge 2 $$, p∈(1,2∗)$$ p\in \left(1,{2}^{\ast}\right) $$, q∈[2,2∗)$$ q\in \left[2,{2}^{\ast}\right) $$, b$$ b $$ is a positive constant, λ≥0$$ \lambda \ge 0 $$ is a parameter, and the potential functions a,V$$ a,V $$ belong to C(ℝN,(0,+∞))$$ C\left({\mathrm{\mathbb{R}}}^N,\left(0,+\infty \right)\right) $$. The existence of radially symmetric and positive solution to the above problem is first established for all λ≥0$$ \lambda \ge 0 $$ when a,V$$ a,V $$ are radially symmetric and p,q∈(4,2∗)$$ p,q\in \left(4,{2}^{\ast}\right) $$, and the range of q$$ q $$ can be extended to q∈(2,2∗)$$ q\in \left(2,{2}^{\ast}\right) $$ with the aid of a coercive type assumption on V$$ V $$. Moreover, we show the existence of infinitely many solutions with high energies via the fountain theorem under more general assumption on V$$ V $$ which allows it to be sign‐changing. When p∈(1,2)$$ p\in \left(1,2\right) $$ and q∈(2,2∗)$$ q\in \left(2,{2}^{\ast}\right) $$, we show that the above problem possesses infinitely many solutions with negative critical values for small λ$$ \lambda $$ provided that the function V−1$$ {V}^{-1} $$ belongs to a suitable space. In particular, by imposing a hypothesis on the potential V$$ V $$ controlling its growth at infinity, we obtain a nonradial solution via the mountain pass theorem and the principle of symmetric criticality. [ABSTRACT FROM AUTHOR]