Planar Schrodinger equations with critical exponential growth

Kód výsledku v 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>

Výsledek na webu

<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>

Jazyk výsledku

angličtina

Název v původním jazyce

Planar Schrodinger equations with critical exponential growth

Popis výsledku v původním jazyce

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.

Název v anglickém jazyce

Planar Schrodinger equations with critical exponential growth

Popis výsledku anglicky

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.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10102 - Applied mathematics

Projekt

—

Návaznosti

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Rok uplatnění

2024

Kód důvěrnosti údajů

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

Název periodika

CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS

ISSN

0944-2669

e-ISSN

1432-0835

Svazek periodika

63

Číslo periodika v rámci svazku

9

Stát vydavatele periodika

US - Spojené státy americké

Počet stran výsledku

46

Strana od-do

„“-„“

Kód UT WoS článku

001352835400001

EID výsledku v databázi Scopus

2-s2.0-85209784997