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Ground state solutions of Nehari-Pohozaev type for Kirchhoff-type problems with general potentials. (English) Zbl 1376.35056

Summary: This paper is dedicated to studying the following Kirchhoff-type problem \[ \begin{cases} -\left( a+b\int _{\mathbb {R}^3}|\nabla u|^2\mathrm {d}x\right) \triangle u+V(x)u=f(u), \quad x\in \mathbb {R}^3; \\ u\in H^1(\mathbb {R}^3), \end{cases} \] where \(a>0,\,b\geq 0\) are two constants, \(V(x)\) is differentiable and \(f\in \mathcal {C}(\mathbb {R}, \mathbb {R})\). By introducing some new tricks, we prove that the above problem admits a ground state solution of Nehari-Pohozaev type and a least energy solution under some mild assumptions on \(V\) and \(f\). Our results generalize and improve the ones in [Z. Guo, J. Differ. Equations 259, No. 7, 2884-2902 (2015; Zbl 1319.35018); G. Li and H. Ye, J. Differ. Equations 257, No. 2, 566–600 (2014; Zbl 1290.35051)] and some other related literature.

MSC:

35J60 Nonlinear elliptic equations
35J20 Variational methods for second-order elliptic equations
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