Ground States of the Nonlinear Schr{\"o}dinger Equation on the Tadpole Graph with a Repulsive Delta Vertex Condition

We consider the stationary nonlinear Schr{\"o}dinger equation set on a tadpole graph with a repulsive delta vertex condition between the loop and the tail of the tadpole. We establish the existence of an action ground state when the size of the loop is either very small or very large. Our analysis relies on variational arguments, such as profile decomposition. When it exists, we study the shape of the ground state using ordinary differential equations arguments, such as the study of period functions. The theoretical results are completed with a numerical study.