Existence of positive ground state solutions for the nonlinear Kirchhoff type equations in \(\mathbb{R}^3\). (English) Zbl 1290.35051
Summary: In this paper, we study the following nonlinear problem of Kirchhoff type with pure power nonlinearities:
\[
\begin{cases} -\left(a + b \int\limits_{\mathbb{R}^3} | D u |^2\right) \operatorname{\Delta} u + V(x) u = | u |^{p - 1} u, & x \in \mathbb{R}^3, \\ u \in H^1\left(\mathbb{R}^3\right), \;u > 0, & x \in \mathbb{R}^3, \end{cases} \quad \eqno {(0.1)}
\]
where \(a, b > 0\) are constants, \(2 < p < 5\) and \(V : \mathbb{R}^3 \to \mathbb{R}\). Under certain assumptions on \(V\), we prove that (0.1) has a positive ground state solution by using a monotonicity trick and a new version of global compactness lemma.
Our main results especially solve problem (0.1) in the case where \(p \in(2, 3]\), which has been an open problem for Kirchhoff equations and can be viewed as a partial extension of a recent result of X. He and W. Zou [J. Differ. Equations 252, No. 2, 1813–1834 (2012; Zbl 1235.35093)] concerning the existence of positive solutions to the nonlinear Kirchhoff problem \[ \begin{cases} -\left(\varepsilon^2 a + \varepsilon b \int\limits_{\mathbb{R}^3} | D u |^2\right) \operatorname{\Delta} u + V(x) u = f(u), & x \in \mathbb{R}^3, \\ u \in H^1(\mathbb{R}^3), u > 0, & x \in \mathbb{R}^3, \end{cases} \] where \(\varepsilon > 0\) is a parameter, \(V(x\)) is a positive continuous potential and \(f(u) \sim | u |^{p - 1} u\) with \(3 < p < 5\) and satisfies the Ambrosetti-Rabinowitz type condition. Our main results extend also the arguments used in [A. Azzollini and A. Pomponio, J. Math. Anal. Appl. 345, No. 1, 90–108 (2008; Zbl 1147.35091); L. Zhao and F. Zhao, J. Math. Anal. Appl. 346, No. 1, 155–169 (2008; Zbl 1159.35017)], which deal with Schrödinger-Poisson system with pure power nonlinearities, to the Kirchhoff type problem.
Our main results especially solve problem (0.1) in the case where \(p \in(2, 3]\), which has been an open problem for Kirchhoff equations and can be viewed as a partial extension of a recent result of X. He and W. Zou [J. Differ. Equations 252, No. 2, 1813–1834 (2012; Zbl 1235.35093)] concerning the existence of positive solutions to the nonlinear Kirchhoff problem \[ \begin{cases} -\left(\varepsilon^2 a + \varepsilon b \int\limits_{\mathbb{R}^3} | D u |^2\right) \operatorname{\Delta} u + V(x) u = f(u), & x \in \mathbb{R}^3, \\ u \in H^1(\mathbb{R}^3), u > 0, & x \in \mathbb{R}^3, \end{cases} \] where \(\varepsilon > 0\) is a parameter, \(V(x\)) is a positive continuous potential and \(f(u) \sim | u |^{p - 1} u\) with \(3 < p < 5\) and satisfies the Ambrosetti-Rabinowitz type condition. Our main results extend also the arguments used in [A. Azzollini and A. Pomponio, J. Math. Anal. Appl. 345, No. 1, 90–108 (2008; Zbl 1147.35091); L. Zhao and F. Zhao, J. Math. Anal. Appl. 346, No. 1, 155–169 (2008; Zbl 1159.35017)], which deal with Schrödinger-Poisson system with pure power nonlinearities, to the Kirchhoff type problem.
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