We study a nonlinear Schrödinger–Poisson system which reduces to the nonlinear and nonlocal PDE -Δu+u+λ2(1ω|x|N-2⋆ρu2)ρ(x)u=|u|q-1ux∈RN,where ω= (N- 2) | SN-1| , λ> 0 , q∈ (1 , 2 ∗- 1) , ρ: RN→ R is nonnegative, locally bounded, and possibly non-radial, N= 3 , 4 , 5 and 2 ∗= 2 N/ (N- 2) is the critical Sobolev exponent. In our setting ρ is allowed as particular scenarios, to either (1) vanish on a region and be finite at infinity, or (2) be large at infinity. We find least energy solutions in both cases, studying the vanishing case by means of a priori integral bounds on the Palais–Smale sequences and highlighting the role of certain positive universal constants for these bounds to hold. Within the Ljusternik–Schnirelman theory we show the existence of infinitely many distinct pairs of high energy solutions, having a min–max characterisation given by means of the Krasnoselskii genus. Our results cover a range of cases where major loss of compactness phenomena may occur, due to the possible unboundedness of the Palais–Smale sequences, and to the action of the group of translations.

Groundstates and infinitely many high energy solutions to a class of nonlinear Schrödinger–Poisson systems / Dutko, T.; Mercuri, C.; Tyler, T. M.. - In: CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS. - ISSN 0944-2669. - 60:5(2021), pp. 1-46. [10.1007/s00526-021-02045-y]

Groundstates and infinitely many high energy solutions to a class of nonlinear Schrödinger–Poisson systems

Mercuri C.;
2021

Abstract

We study a nonlinear Schrödinger–Poisson system which reduces to the nonlinear and nonlocal PDE -Δu+u+λ2(1ω|x|N-2⋆ρu2)ρ(x)u=|u|q-1ux∈RN,where ω= (N- 2) | SN-1| , λ> 0 , q∈ (1 , 2 ∗- 1) , ρ: RN→ R is nonnegative, locally bounded, and possibly non-radial, N= 3 , 4 , 5 and 2 ∗= 2 N/ (N- 2) is the critical Sobolev exponent. In our setting ρ is allowed as particular scenarios, to either (1) vanish on a region and be finite at infinity, or (2) be large at infinity. We find least energy solutions in both cases, studying the vanishing case by means of a priori integral bounds on the Palais–Smale sequences and highlighting the role of certain positive universal constants for these bounds to hold. Within the Ljusternik–Schnirelman theory we show the existence of infinitely many distinct pairs of high energy solutions, having a min–max characterisation given by means of the Krasnoselskii genus. Our results cover a range of cases where major loss of compactness phenomena may occur, due to the possible unboundedness of the Palais–Smale sequences, and to the action of the group of translations.
2021
60
5
1
46
Groundstates and infinitely many high energy solutions to a class of nonlinear Schrödinger–Poisson systems / Dutko, T.; Mercuri, C.; Tyler, T. M.. - In: CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS. - ISSN 0944-2669. - 60:5(2021), pp. 1-46. [10.1007/s00526-021-02045-y]
Dutko, T.; Mercuri, C.; Tyler, T. M.
File in questo prodotto:
Non ci sono file associati a questo prodotto.
Pubblicazioni consigliate

Licenza Creative Commons
I metadati presenti in IRIS UNIMORE sono rilasciati con licenza Creative Commons CC0 1.0 Universal, mentre i file delle pubblicazioni sono rilasciati con licenza Attribuzione 4.0 Internazionale (CC BY 4.0), salvo diversa indicazione.
In caso di violazione di copyright, contattare Supporto Iris

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11380/1292206
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 3
  • ???jsp.display-item.citation.isi??? 3
social impact