Inhomogeneous nonlinear Schr\"odinger equations with competing singular nonlinearities

TL;DR

This paper investigates stationary solutions of inhomogeneous nonlinear Schr"odinger equations with competing singular nonlinearities, establishing existence, multiplicity, and nonexistence results through variational methods and weighted inequalities.

Contribution

It introduces a new variational framework for these equations using weighted Sobolev spaces and derives multiple existence and multiplicity results, including radial improvements.

Findings

Existence of a sequence of nonlinear eigenvalues via min-max methods.

Broad conditions for existence and multiplicity in subcritical and critical cases.

Nonexistence results based on Pohozaev-type identities.

Abstract

We study nonlinear elliptic equations that arise as stationary states of inhomogeneous nonlinear Schr\"odinger equations with competing singular nonlinearities. The model involves the Laplacian combined with weighted power-type terms and naturally leads to a variational formulation in a weighted Sobolev space obtained from the intersection of the homogeneous Sobolev space with a weighted Lebesgue space. Using sharp weighted Sobolev and Caffarelli--Kohn--Nirenberg type inequalities, we establish continuous and compact embeddings of this space into suitable weighted Lebesgue spaces. These embedding results, together with a natural scaling structure of the model, allow us to apply the abstract critical point framework of Mercuri and Perera (2026), yielding a sequence of nonlinear eigenvalues for the associated problem via a min--max scheme based on the Fadell--Rabinowitz cohomological index. Within this framework we obtain a broad collection of existence and multiplicity results for equations driven by sums of weighted power nonlinearities, covering interactions in both subcritical and critical cases. We also establish a nonexistence result derived from a Pohozaev-type identity. Finally, we analyze the radial setting, where improved radial Caffarelli--Kohn--Nirenberg inequalities allow us to enlarge some of the admissible embedding ranges. This leads to strengthened radial versions of our main results.