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.