Semi-classical solutions for Kirchhoff type problem with a critical frequency. (English) Zbl 1465.35219
Summary: In the present paper, we consider the following Kirchhoff type problem
\[
-\Big(\varepsilon^2+\varepsilon b\int_{\mathbb{R}^3}|\nabla v|^2\Big)\Delta v+V(x)v=|v|^{p-2}v\quad\text{in }\mathbb{R}^3,
\]
where \(b > 0\), \(p\in(4,6)\), the potential \(V\in C(\mathbb{R}^3,\mathbb{R})\) and \(\varepsilon\) is a positive parameter. The existence and multiplicity of semi-classical state solutions are obtained by variational method for this problem with several classes of critical frequency potentials, i.e., \( \inf_{\mathbb{R}^N} V=0\). As to Kirchhoff type problem, little has been done for the critical frequency cases in the literature, especially the potential may vanish at infinity.
MSC:
| 35J60 | Nonlinear elliptic equations |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
| 35A15 | Variational methods applied to PDEs |
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