We study the existence of ground states to a nonlinear fractional Kirchhoff equation with an external potential V . Under suitable assumptions on V , using the monotonicity trick and the profile decomposition, we prove the existence of ground states. In particular, the nonlinearity does not satisfy the Ambrosetti-Rabinowitz type condition or monotonicity assumptions.

In this article, we consider the multiplicity of positive solutions for a class of Kirchhoff type problems with concave and convex nonlinearities. Under appropriate assumptions, we prove that the problem has at least two positive solutions, moreover, one of which is a positive ground state solution. Our approach is mainly based on the Nehari manifold, Ekeland variational principle and the theor...

We show under mild assumptions that a composition of internally well-posed, impedance passive (or conservative) boundary control systems through Kirchhoff type connections is also an internally well-posed, impedance passive (resp., conservative) boundary control system. The proof is based on results of [21]. We also present an example of such composition involving Webster’s equation on a Y-shap...

We compute explicitly the solution of the heat equation on a weighted graph 0 whose edges are identified with copies of the segment [0, 1] with the condition that the sum of the weighted normal exterior derivatives is 0 at every node (Kirchhoff type condition).