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Inhomogeneous nonlinear Schrödinger equations with competing singular nonlinearities

Elisandra Gloss, Kanishka Perera, Bruno Ribeiro

Abstract

We study nonlinear elliptic equations arising as stationary states of inhomogeneous nonlinear Schrödinger equations with competing singular nonlinearities. Working in a weighted Sobolev space that combines the homogeneous Sobolev space with a weighted Lebesgue term, we establish continuous and compact embeddings of Caffarelli--Kohn--Nirenberg type. These embeddings, together with a model that displays a natural scaling, allow us to apply the abstract critical point framework of Mercuri and Perera (2025), yielding a sequence of nonlinear eigenvalues for the associated problem. This scaling property leads to a classification of weighted power-type nonlinearities into subscaled, scaled, and superscaled regimes. Within this variational setting, we obtain broad existence and multiplicity results for equations driven by sums of weighted power nonlinearities, covering superscaled, scaled, and subscaled interactions, both in the subcritical and critical cases. We also provide a nonexistence result as a consequence of a Pohozaev-type identity. Finally, in the radial setting we employ improved radial CKN inequalities to enlarge the admissible embedding ranges. This yields strengthened radial versions of all our main results, including two-dimensional configurations with more singular weights, where no compact embeddings are available in the nonradial case.

Inhomogeneous nonlinear Schrödinger equations with competing singular nonlinearities

Abstract

We study nonlinear elliptic equations arising as stationary states of inhomogeneous nonlinear Schrödinger equations with competing singular nonlinearities. Working in a weighted Sobolev space that combines the homogeneous Sobolev space with a weighted Lebesgue term, we establish continuous and compact embeddings of Caffarelli--Kohn--Nirenberg type. These embeddings, together with a model that displays a natural scaling, allow us to apply the abstract critical point framework of Mercuri and Perera (2025), yielding a sequence of nonlinear eigenvalues for the associated problem. This scaling property leads to a classification of weighted power-type nonlinearities into subscaled, scaled, and superscaled regimes. Within this variational setting, we obtain broad existence and multiplicity results for equations driven by sums of weighted power nonlinearities, covering superscaled, scaled, and subscaled interactions, both in the subcritical and critical cases. We also provide a nonexistence result as a consequence of a Pohozaev-type identity. Finally, in the radial setting we employ improved radial CKN inequalities to enlarge the admissible embedding ranges. This yields strengthened radial versions of all our main results, including two-dimensional configurations with more singular weights, where no compact embeddings are available in the nonradial case.
Paper Structure (15 sections, 28 theorems, 283 equations)

This paper contains 15 sections, 28 theorems, 283 equations.

Table of Contents

  1. Introduction
  2. Variational framework
  3. Main theorems
  4. Preliminaries
  5. Embeddings in weighted Lebesgue spaces
  6. Pohozaev Identity and nonexistence results
  7. Scaled operators on $E^q_b(\mathbb R^N)$
  8. Scaled eigenvalue problem
  9. Proof of the main theorems
  10. On the Palais-Smale condition
  11. Existence of solutions for subcritical superscaled problems
  12. Existence and multiplicity for critical problems
  13. Problems with subscaled nonlinearities
  14. Radial improvements
  15. Appendix

Key Result

Theorem 1.1

Assume basicparametershypothesis. Suppose $f(x,t)=\lambda|x|^{-a}|t|^{p-2}t+|x|^{-\eta}h(t)$, with $\lambda\in\mathbb{R}$, $0<\eta<N-q(N-2)/2$ ($\eta\in(b,2)$ if $N=2$) and $h=H'$ for $H:\mathbb{R}\to\mathbb{R}$ a $C^1$ function such that for some constants $c,c_1,c_2>0$, and superscaled pairs $(\eta,r_1)$, $(\eta,r_2)$ and $(\eta,r)$, with $2^*_{b,q,\eta}<r_1,r_2<2^*_\eta$ and $q<r<2^*_\eta$. Th

Theorems & Definitions (47)

  • Definition 1
  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Theorem 1.9
  • ...and 37 more