TY - JOUR
T1 - On a Class of Quasilinear Elliptic Equations
AU - Hashimi , Sayed Hamid
AU - Wang , Zhi-Qiang
AU - Zhang , Lin
JO - Communications in Mathematical Research 
VL - 2
SP - 209
EP - 230
PY - 2023
DA - 2023/04
SN - 39
DO - http://doi.org/10.4208/cmr.2022-0038
UR - https://global-sci.org/intro/article_detail/cmr/21545.html
KW - Variational perturbations, $p$-Laplacian regularization, quasilinear elliptic equations, modified nonlinear Schrödinger equations, sign-changing solutions, critical exponents.
AB - <p style="text-align: justify;">We consider a class of quasilinear elliptic boundary problems, including the following Modified Nonlinear Schrödinger Equation as a special case: $$\begin{cases} ∆u+ \frac{1}{2} u∆(u^2)−V(x)u+|u|^{q−2}u=0 \ \ \ in \ Ω, \\u=0 \ \ \ \ \ \ \&nbsp; ~ ~ ~ on \&nbsp; ∂Ω, \end{cases}$$ 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.</p>