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Volume 39, Issue 2
On a Class of Quasilinear Elliptic Equations

Sayed Hamid Hashimi, Zhi-Qiang Wang & Lin Zhang

DOI: 10.4208/cmr.2022-0038

Commun. Math. Res., 39 (2023), pp. 209-230.

Published online: 2023-04

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  • 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.

  • Keywords

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

  • AMS Subject Headings

35B08, 35B20, 35B33, 35J20, 35J62, 35J92, 35Q55, 35P30

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CMR-39-209, author = {Hashimi , Sayed HamidWang , Zhi-Qiang and Zhang , Lin}, title = {On a Class of Quasilinear Elliptic Equations}, journal = {Communications in Mathematical Research }, year = {2023}, volume = {39}, number = {2}, pages = {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.

}, issn = {2707-8523}, doi = {https://doi.org/10.4208/cmr.2022-0038}, url = {http://global-sci.org/intro/article_detail/cmr/21545.html} }
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 -

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.

Hashimi , Sayed HamidWang , Zhi-Qiang and Zhang , Lin. (2023). On a Class of Quasilinear Elliptic Equations. Communications in Mathematical Research . 39 (2). 209-230. doi:10.4208/cmr.2022-0038
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