Planar Schrodinger equations with critical exponential growth

Result code in IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216305%3A26220%2F24%3APU152705" target="_blank" >RIV/00216305:26220/24:PU152705 - isvavai.cz</a>

Result on the web

<a href="https://doi.org/10.1007/s00526-024-02852-z" target="_blank" >https://doi.org/10.1007/s00526-024-02852-z</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1007/s00526-024-02852-z" target="_blank" >10.1007/s00526-024-02852-z</a>

Result language

angličtina

Original language name

Planar Schrodinger equations with critical exponential growth

Original language description

In this paper, we study the following quasilinear Schrodinger equation: -epsilon(2)Delta u + V(x)u - epsilon(2)Delta(u(2))u = g(u), x is an element of R-2, where epsilon > 0 is a small parameter, V is an element of C(R-2, R) is uniformly positive and allowed to be unbounded from above, and g is an element of C(R, R) has a critical exponential growth at infinity. In the autonomous case, when epsilon > 0 is fixed and V(x) equivalent to V-0 is an element of R+, we first present a remarkable relationship between the existence of least energy solutions and the range of V-0 without any monotonicity conditions on g. Based on some new strategies, we establish the existence and concentration of positive solutions for the above singularly perturbed problem. In particular, our approach not only permits to extend the previous results to a wider class of potentials V and source terms g, but also allows a uniform treatment of two kinds of representative nonlinearities that g has extra restrictions at infinity or near the origin, namely lim inf(|t|->+infinity)tg(t)/e(0)(alpha)t4 or g(u) >= C-q,C-V u(q-1) with q > 4 and C-q,C- V > 0 is an implicit value depending on q, V and the best constant of the embedding H-1(R-2) subset of L-q(R-2), considered in the existing literature. To the best of our knowledge, there have not been established any similar results, even for simpler semilinear Schrodinger equations. We believe that our approach could be adopted and modified to treat more general elliptic partial differential equations involving critical exponential growth.

Czech name

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Czech description

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Type

J_imp - Article in a specialist periodical, which is included in the Web of Science database

CEP classification

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OECD FORD branch

10102 - Applied mathematics

Project

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Continuities

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Publication year

2024

Confidentiality

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Name of the periodical

CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS

ISSN

0944-2669

e-ISSN

1432-0835

Volume of the periodical

63

Issue of the periodical within the volume

9

Country of publishing house

US - UNITED STATES

Number of pages

46

Pages from-to

„“-„“

UT code for WoS article

001352835400001

EID of the result in the Scopus database

2-s2.0-85209784997