Least energy sign-changing solutions for Kirchhoff-Poisson systems. (English) Zbl 1524.35220
Summary: The paper deals with the following Kirchhoff-Poisson systems:
\[
\begin{cases}
- ({1+b\int_{{\mathbb{R}}^3} { \vert \nabla u \vert^2\,dx}}) \Delta u+u+k(x)\phi u+\lambda \vert u \vert^{p-2}u=h(x) \vert u \vert^{q-2}u, & x\in{\mathbb{R}}^3, \\
-\Delta \phi =k(x)u^2, & x\in{\mathbb{R}}^3,
\end{cases}
\] where the functions \(k\) and \(h\) are nonnegative, \(0\le \lambda\), \(b; 2\le p\le 4< q<6\). Via a constraint variational method combined with a quantitative lemma, some existence results on one least energy sign-changing solution with two nodal domains to the above systems are obtained. Moreover, the convergence property of \(u_b\) as \(b \searrow 0\) is established.
MSC:
| 35J47 | Second-order elliptic systems |
| 35J50 | Variational methods for elliptic systems |
| 35J60 | Nonlinear elliptic equations |
Keywords:
Kirchhoff-Poisson systems; least energy sign-changing solutions; constraint variational method; nodal domainsReferences:
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