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Existence and Qualitative Properties of Solutions to Nonlinear Schrodinger-Poisson Systems / TERESA TYLER

Swansea University Author: TERESA TYLER

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DOI (Published version): 10.23889/SUthesis.56842

Abstract

Our main equation of study is the nonlinear Schr¨odinger-Poisson system⇢−Du+u+r(x)fu = |u|p−1u, x 2 R3,−Df = r(x)u2, x 2 R3,with p 2 (2,5) and r : R3 ! R a nonnegative measurable function. In the spirit ofthe classical work of P. H. Rabinowitz [55] on nonlinear Schr¨odinger equations, wefirst prove...

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Published: Swansea 2020
Institution: Swansea University
Degree level: Doctoral
Degree name: Ph.D
Supervisor: Mercuri, Carlo ; Crooks, Elaine
URI: https://cronfa.swan.ac.uk/Record/cronfa56842
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spelling 2021-05-10T16:16:17.6383775 v2 56842 2021-05-10 Existence and Qualitative Properties of Solutions to Nonlinear Schrodinger-Poisson Systems 8a7b969770fb33e661c2c7e4ab06482a TERESA TYLER TERESA TYLER true false 2021-05-10 Our main equation of study is the nonlinear Schr¨odinger-Poisson system⇢−Du+u+r(x)fu = |u|p−1u, x 2 R3,−Df = r(x)u2, x 2 R3,with p 2 (2,5) and r : R3 ! R a nonnegative measurable function. In the spirit ofthe classical work of P. H. Rabinowitz [55] on nonlinear Schr¨odinger equations, wefirst prove existence of positive mountain-pass solutions and least energy solutions tothis system under different assumptions on r at infinity. Our results cover the rangep 2 (2,3) where the lack of compactness phenomena may be due to the combinedeffect of the invariance by translations of a ‘limiting problem’ at infinity and of thepossible unboundedness of the Palais-Smale sequences. In the case of a coercive r,namely r(x)!+• as |x|!+•, we then prove the existence of infinitely many distinctpairs of solutions. For p 2 (3,5) we exploit the symmetry of the problem bythe action of Z2 as well as some well-known properties of the Krasnoselskii-genus,whereas for p 2 (2,3] we use an appropriate abstract min-max scheme, which requiressome additional assumptions on r.After establishing these existence and multiplicity results, we are then interested inthe qualitative properties of solutions the singularly perturbed problem⇢−e2Du+lu+r(x)fu = |u|p−1u, x 2 R3−Df = r(x)u2, x 2 R3,with r : R3 ! R a nonnegative measurable function, l 2 R, and l > 0, taking advantageof a shrinking parameter e ⌧ 1. In particular, we seek to understand theconcentration phenomena purely driven by r. To this end, we first find necessaryconditions for concentration at points to occur for solutions in various functionalsettings which are suitable for both variational and perturbation methods. We thendiscuss a variational/penalisation method, which has been exploited in the case ofnonlinear Schr¨odinger equations, and discuss its applications to the present nonlinearSchr¨odinger-Poisson context, in the attempt of showing that the necessary conditionsare, in fact, sufficient conditions on r for point concentration of solutions. Finally,we present some preliminary results in this direction that elicit interesting standalonequalitative properties of the solutions. E-Thesis Swansea 15 4 2020 2020-04-15 10.23889/SUthesis.56842 COLLEGE NANME COLLEGE CODE Swansea University Mercuri, Carlo ; Crooks, Elaine Doctoral Ph.D 2021-05-10T16:16:17.6383775 2021-05-10T15:21:23.6828291 Faculty of Science and Engineering School of Mathematics and Computer Science - Mathematics TERESA TYLER 1 Under embargo Under embargo 2021-05-10T15:33:44.4826999 Output 1330765 application/pdf E-Thesis – open access true 2023-05-01T00:00:00.0000000 Copyright: The author, Teresa Megan Tyler, 2020. true eng
title Existence and Qualitative Properties of Solutions to Nonlinear Schrodinger-Poisson Systems
spellingShingle Existence and Qualitative Properties of Solutions to Nonlinear Schrodinger-Poisson Systems
TERESA TYLER
title_short Existence and Qualitative Properties of Solutions to Nonlinear Schrodinger-Poisson Systems
title_full Existence and Qualitative Properties of Solutions to Nonlinear Schrodinger-Poisson Systems
title_fullStr Existence and Qualitative Properties of Solutions to Nonlinear Schrodinger-Poisson Systems
title_full_unstemmed Existence and Qualitative Properties of Solutions to Nonlinear Schrodinger-Poisson Systems
title_sort Existence and Qualitative Properties of Solutions to Nonlinear Schrodinger-Poisson Systems
author_id_str_mv 8a7b969770fb33e661c2c7e4ab06482a
author_id_fullname_str_mv 8a7b969770fb33e661c2c7e4ab06482a_***_TERESA TYLER
author TERESA TYLER
author2 TERESA TYLER
format E-Thesis
publishDate 2020
institution Swansea University
doi_str_mv 10.23889/SUthesis.56842
college_str Faculty of Science and Engineering
hierarchytype
hierarchy_top_id facultyofscienceandengineering
hierarchy_top_title Faculty of Science and Engineering
hierarchy_parent_id facultyofscienceandengineering
hierarchy_parent_title Faculty of Science and Engineering
department_str School of Mathematics and Computer Science - Mathematics{{{_:::_}}}Faculty of Science and Engineering{{{_:::_}}}School of Mathematics and Computer Science - Mathematics
document_store_str 0
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description Our main equation of study is the nonlinear Schr¨odinger-Poisson system⇢−Du+u+r(x)fu = |u|p−1u, x 2 R3,−Df = r(x)u2, x 2 R3,with p 2 (2,5) and r : R3 ! R a nonnegative measurable function. In the spirit ofthe classical work of P. H. Rabinowitz [55] on nonlinear Schr¨odinger equations, wefirst prove existence of positive mountain-pass solutions and least energy solutions tothis system under different assumptions on r at infinity. Our results cover the rangep 2 (2,3) where the lack of compactness phenomena may be due to the combinedeffect of the invariance by translations of a ‘limiting problem’ at infinity and of thepossible unboundedness of the Palais-Smale sequences. In the case of a coercive r,namely r(x)!+• as |x|!+•, we then prove the existence of infinitely many distinctpairs of solutions. For p 2 (3,5) we exploit the symmetry of the problem bythe action of Z2 as well as some well-known properties of the Krasnoselskii-genus,whereas for p 2 (2,3] we use an appropriate abstract min-max scheme, which requiressome additional assumptions on r.After establishing these existence and multiplicity results, we are then interested inthe qualitative properties of solutions the singularly perturbed problem⇢−e2Du+lu+r(x)fu = |u|p−1u, x 2 R3−Df = r(x)u2, x 2 R3,with r : R3 ! R a nonnegative measurable function, l 2 R, and l > 0, taking advantageof a shrinking parameter e ⌧ 1. In particular, we seek to understand theconcentration phenomena purely driven by r. To this end, we first find necessaryconditions for concentration at points to occur for solutions in various functionalsettings which are suitable for both variational and perturbation methods. We thendiscuss a variational/penalisation method, which has been exploited in the case ofnonlinear Schr¨odinger equations, and discuss its applications to the present nonlinearSchr¨odinger-Poisson context, in the attempt of showing that the necessary conditionsare, in fact, sufficient conditions on r for point concentration of solutions. Finally,we present some preliminary results in this direction that elicit interesting standalonequalitative properties of the solutions.
published_date 2020-04-15T04:12:06Z
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score 11.012678

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