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Point interactions and singular solutions to semilinear elliptic equations

Filippo Boni, Diego Noja, Raffaele Scandone

Abstract

We investigate the connection between semilinear elliptic PDEs with isolated singularities and stationary nonlinear Schrödinger equations with point interactions. In dimensions $d=2,3$, we provide a detailed equivalence result between the two problems. As a consequence, this allows us to exploit a range of operator-theoretic and variational techniques, hitherto not explicitly explored in the context of singular solutions. By leveraging this approach, in the focusing case, we provide the existence of infinitely many singular solutions by applying the Ambrosetti-Rabinowitz theory to an action functional adapted to the point interaction. When $d=2$ and relying on a suitable uniqueness result, we also characterize positive solutions in terms of action ground states, and we show the existence of infinitely many singular, nodal solutions.

Point interactions and singular solutions to semilinear elliptic equations

Abstract

We investigate the connection between semilinear elliptic PDEs with isolated singularities and stationary nonlinear Schrödinger equations with point interactions. In dimensions , we provide a detailed equivalence result between the two problems. As a consequence, this allows us to exploit a range of operator-theoretic and variational techniques, hitherto not explicitly explored in the context of singular solutions. By leveraging this approach, in the focusing case, we provide the existence of infinitely many singular solutions by applying the Ambrosetti-Rabinowitz theory to an action functional adapted to the point interaction. When and relying on a suitable uniqueness result, we also characterize positive solutions in terms of action ground states, and we show the existence of infinitely many singular, nodal solutions.
Paper Structure (17 sections, 14 theorems, 100 equations)

This paper contains 17 sections, 14 theorems, 100 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Point interactions and the equivalence
  4. Variational structure and critical points of the action
  5. Positive versus nodal solutions in R2
  6. Proofs of Main Results
  7. Equivalence
  8. Existence of critical points of the action
  9. Structure of positive solutions and existence of nodal solutions
  10. Further extensions and remarks
  11. Other nonlinearities and generalizations
  12. The absorption case
  13. Complex solutions
  14. Time-dependent problems
  15. Other approaches
  16. ...and 2 more sections

Key Result

Theorem 2.1

Let $\lambda>0$, $\sigma=\pm 1$, and either $d=2$, $p\in(1,\infty)$ or $d=3$, $p\in(1,3)$. The following conditions are equivalent: If the above conditions are satisfied, then and $f:=u-q\mathcal{G}_{\lambda}\in H^s({\mathbb R}^d)$. Moreover, $u$ extends to a regular solution (i.e. of class $\mathcal{C}^2$ on ${\mathbb R}^d$) if and only if $q=0$: in this case, $u\in H^2_{\infty}({\mathbb R}^d)=

Theorems & Definitions (28)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Remark 2.5
  • Proposition 3.1
  • Remark 3.2
  • Lemma 3.3
  • proof
  • Proposition 3.4: Brezis-Lions Lemma
  • ...and 18 more