Nodal solutions for a Kirchhoff type problem in \(\mathbb{R}^N\). (English) Zbl 1411.35122
Summary: In this paper, we consider the existence of nodal solutions of the following Kirchhoff problem
\[-\left(a+b\int_{\mathbb R^N}\vert\nabla u\vert^2 dx\right)\Delta u +u=g(u),\quad\text{in }\mathbb R^N, \tag{\(\mathcal{KP}\)}\]
where \(a, b > 0\) and \(N \geq 3\). We first prove that \((\mathcal{KP})\) is equivalent to the following system with respect to \((u, \lambda)\):
\[ \begin{cases} - \Delta u + u = g(u), \\ \lambda - a - b \lambda^{\frac{N - 2}{2}} \int_{\mathbb R^N} | \nabla u |^2 d x = 0, \end{cases} \quad \text{in } \mathbb R^N \times \mathbb R^+ .\tag{\(\mathcal F_\lambda\)}\]
For every integer \(k > 0\), radial solutions of \((\mathcal{KP})\) with exactly \(k\) nodes are obtained by dealing with the system \((\mathcal{F}_\lambda)\) under some suitable assumptions. Moreover, nonexistence results are proved if \(N \geq 4\) and \(b\) is sufficiently large.
\[-\left(a+b\int_{\mathbb R^N}\vert\nabla u\vert^2 dx\right)\Delta u +u=g(u),\quad\text{in }\mathbb R^N, \tag{\(\mathcal{KP}\)}\]
where \(a, b > 0\) and \(N \geq 3\). We first prove that \((\mathcal{KP})\) is equivalent to the following system with respect to \((u, \lambda)\):
\[ \begin{cases} - \Delta u + u = g(u), \\ \lambda - a - b \lambda^{\frac{N - 2}{2}} \int_{\mathbb R^N} | \nabla u |^2 d x = 0, \end{cases} \quad \text{in } \mathbb R^N \times \mathbb R^+ .\tag{\(\mathcal F_\lambda\)}\]
For every integer \(k > 0\), radial solutions of \((\mathcal{KP})\) with exactly \(k\) nodes are obtained by dealing with the system \((\mathcal{F}_\lambda)\) under some suitable assumptions. Moreover, nonexistence results are proved if \(N \geq 4\) and \(b\) is sufficiently large.
MSC:
| 35J60 | Nonlinear elliptic equations |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
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