## Existence of positive ground state solutions for Kirchhoff type problems.(English)Zbl 1318.35121

Summary: In this paper, we study the existence of positive ground state solutions for the nonlinear Kirchhoff type problem $\begin{cases} -\left(a + b \int_{\mathbb R^3}| \nabla u |^2\right) \triangle u + V(x) u = f(u) & \text{in } \mathbb R^3, \\ u \in H^1(\mathbb R^3), u > 0 & \text{in } \mathbb R^3, \end{cases}$ where $$a, b > 0$$ are constants, $$f \in C(\mathbb R, \mathbb R)$$ is subcritical near infinity and superlinear near zero and satisfies the Berestycki-Lions condition. By using an abstract critical point theorem established by Jeanjean and a new global compactness lemma, we show that the above problem has at least a positive ground state solution. Our result generalizes the results of G. Li and H. Ye [J. Differ. Equations 257, No. 2, 566–600 (2014; Zbl 1290.35051)] concerning the nonlinearity $$f(u) = | u |^{p - 1} u$$ with $$p \in (2, 5)$$.

### MSC:

 35Q74 PDEs in connection with mechanics of deformable solids 35Q51 Soliton equations 53C35 Differential geometry of symmetric spaces 35B09 Positive solutions to PDEs 35R09 Integro-partial differential equations 74K05 Strings 74K15 Membranes 74H45 Vibrations in dynamical problems in solid mechanics

Zbl 1290.35051
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