Infinitely Many Normalized Solutions for the Kirchhoff Equation with Localized Nonlinearities on Noncompact Metric Graphs
Wangwang Peng *
School of Mathematics and Statistics, Southwest University, Chongqing, 400715, P.R. China.
*Author to whom correspondence should be addressed.
Abstract
In this paper, we study the following Kirchhoff problem
\(\left\{\begin{array}{l}
-\left(a+b \int_{\mathcal{G}}\left|u^{\prime}\right|^2 d x\right) u^{\prime \prime}+\lambda u=\kappa(x)|u|^{p-2} u \quad x \in \mathcal{G} \\
\int_{\mathcal{G}}|u|^2 d x=\mu
\end{array}\right.\)
where p > 10, a, b > 0, μ > 0 is a prescribed mass, G is a noncompact metric graph, κ is the characteristic function of the compact core K of G and λ ∈ \(\mathbb{R}\) appears as a Lagrange multiplier. By using minimax principle, we prove the existence of infinitely many normalized solutions for any prescribed mass in the L2-supercritical case.
Keywords: Kirchhoff equation, L2-supercritical, infinitely many solutions, noncompact metric graph