Publication detail

Planar Schrodinger equations with critical exponential growth

CHEN, S. RADULESCU, V. TANG, X. WEN, L.

Original Title

Planar Schrodinger equations with critical exponential growth

Type

journal article in Web of Science

Language

English

Original Abstract

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.

Keywords

LINEAR ELLIPTIC-EQUATIONS;GROUND-STATE SOLUTION;SOLITON-SOLUTIONS;NONTRIVIAL SOLUTION;POSITIVE SOLUTIONS;EXISTENCE;INEQUALITIES;R-2

Authors

CHEN, S.; RADULESCU, V.; TANG, X.; WEN, L.

Released

11. 12. 2024

ISBN

0944-2669

Periodical

CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS

Year of study

63

Number

9

State

United States of America

Pages count

46

URL

https://doi.org/10.1007/s00526-024-02852-z

BibTex

@article{BUT191299,
  author="Sitong {Chen} and Vicentiu {Radulescu} and Xianhua {Tang} and Lixi {Wen}",
  title="Planar Schrodinger equations with critical exponential growth
",
  journal="CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS",
  year="2024",
  volume="63",
  number="9",
  pages="46",
  doi="10.1007/s00526-024-02852-z",
  issn="0944-2669",
  url="https://doi.org/10.1007/s00526-024-02852-z"
}