On a Class of Quasilinear Elliptic Equations

On a Class of Quasilinear Elliptic Equations

Year:    2023

Author:    Sayed Hamid Hashimi, Zhi-Qiang Wang, Lin Zhang

Communications in Mathematical Research , Vol. 39 (2023), Iss. 2 : pp. 209–230

Abstract

We consider a class of quasilinear elliptic boundary problems, including the following Modified Nonlinear Schrödinger Equation as a special case: $$\begin{cases} ∆u+ \frac{1}{2} u∆(u^2)−V(x)u+|u|^{q−2}u=0 \ \ \ in \ Ω, \\u=0 \ \ \ \ \ \ \  ~ ~ ~ on \  ∂Ω, \end{cases}$$ where $Ω$ is the entire space $\mathbb{R}^N$ or $Ω ⊂ \mathbb{R}^N$ is a bounded domain with smooth boundary, $q∈(2,22^∗]$ with $2^∗=2N/(N−2)$ being the critical Sobolev exponent and $22^∗= 4N/(N−2).$ We review the general methods developed in the last twenty years or so for the studies of existence, multiplicity, nodal property of the solutions within this range of nonlinearity up to the new critical exponent $4N/(N−2),$ which is a unique feature for this class of problems. We also discuss some related and more general problems.

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Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/cmr.2022-0038

Communications in Mathematical Research , Vol. 39 (2023), Iss. 2 : pp. 209–230

Published online:    2023-01

AMS Subject Headings:    Global Science Press

Copyright:    COPYRIGHT: © Global Science Press

Pages:    22

Keywords:    Variational perturbations $p$-Laplacian regularization quasilinear elliptic equations modified nonlinear Schrödinger equations sign-changing solutions critical exponents.

Author Details

Sayed Hamid Hashimi

Zhi-Qiang Wang

Lin Zhang