Bounded state solutions of Kirchhoff type problems with a critical exponent in high dimension. (English) Zbl 1401.35090
Summary: In the present paper, we consider the following Kirchhoff type problem
\[
\begin{cases} -\left(a+\lambda\int_{\mathbb{R}^N} |\nabla u|^2dx\right) \Delta u+V(x)u = |u|^{2^*-2}u \;\;\text{ in } \mathbb{R}^N, \\ u \in D^{1,2}(\mathbb{R}^N),\end{cases}
\]
where \(a\) is a positive constant, \(\lambda\) is a positive parameter, \(V\in L^{\frac{N}{2}}(\mathbb{R}^N)\) is a given nonnegative function and \(2^*\) is the critical exponent. The existence of bounded state solutions for Kirchhoff type problem with critical exponents in the whole \(\mathbb{R}^N\) (\(N \geq 5\)) has never been considered so far. We obtain sufficient conditions on the existence of bounded state solutions in high dimension \(N \geq 4\), and especially it is the fist time to consider the case when \(N \geq 5\) in the literature.
MSC:
| 35J60 | Nonlinear elliptic equations |
| 47J30 | Variational methods involving nonlinear operators |
| 35J20 | Variational methods for second-order elliptic equations |
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