Normalized multi-bump solutions of nonlinear Kirchhoff equations

Abstract

We are concerned with the existence and concentration of multi-bump solutions for the nonlinear Kirchhoff equation $ \begin{eqnarray*} -\left ( \varepsilon ^{2}a+\varepsilon b\displaystyle {\int}_{\mathbb{R}^{3} }\left | \nabla v \right | ^{2} \mathrm {d} x \right )\Delta v+\lambda v = K(x)\left | v \right |^{2\sigma }v,\,\,\,x\in\mathbb{R}^3 \end{eqnarray*} $ with an $ L^{2} $-constraint in the $ L^{2} $-subcritical case $ \sigma\in\left(0, \, \frac{2}{3}\right) $ and the $ L^{2} $-supercritical case $ \sigma\in\left(\frac{2}{3}, \, 2 \right). $ Here $ \lambda \in \mathbb{R} $ appears as a Lagrange multiplier, $ \varepsilon $ is a small positive parameter and $ K > 0 $ possesses several local maximum points. By employing the variational gluing method and the penalization technique, we prove the existence of multi-bump solutions that are concentrated at local maximum points of $ K $ for the problem above.

Keywords

• kirchhoff equation
• normalized solutions
• concentration behavior