Existence and asymptotic behavior of standing wave solutions for a class of generalized quasilinear Schrödinger equations with critical Sobolev exponents

2020 ◽  
Vol 120 (3-4) ◽  
pp. 199-248
Author(s):  
Jianhua Chen ◽  
Xianjiu Huang ◽  
Dongdong Qin ◽  
Bitao Cheng
Keyword(s):  
Asymptotic Behavior ◽  
Ground State ◽  
Multiple Solutions ◽  
Global Minimum ◽  
Global Maximum ◽  
Quasilinear Schrödinger Equation ◽  
Critical Sobolev Exponents ◽  
Wave Solutions ◽  
Change Of Variable ◽  
Concentration Behavior

In this paper, we study the following generalized quasilinear Schrödinger equation − div ( ε 2 g 2 ( u ) ∇ u ) + ε 2 g ( u ) g ′ ( u ) | ∇ u | 2 + V ( x ) u = K ( x ) | u | p − 2 u + | u | 22 ∗ − 2 u , x ∈ R N , where N ⩾ 3, ε > 0, 4 < p < 22 ∗ , g ∈ C 1 ( R , R + ), V ∈ C ( R N ) ∩ L ∞ ( R N ) has a positive global minimum, and K ∈ C ( R N ) ∩ L ∞ ( R N ) has a positive global maximum. By using a change of variable, we obtain the existence and concentration behavior of ground state solutions for this problem with critical growth, and establish a phenomenon of exponential decay. Moreover, by Ljusternik–Schnirelmann theory, we also prove the existence of multiple solutions.

scholarly journals Existence and concentration of positive solutions for a p-fractional Choquard equation

2021 ◽  
Vol 6 (11) ◽  
pp. 12929-12951
Author(s):  
Xudong Shang ◽  
Keyword(s):  
Small Parameter ◽  
Positive Solutions ◽  
Ground State ◽  
Global Minimum ◽  
Ground State Solution ◽  
Global Maximum ◽  
Choquard Equation ◽  
Number Of Solutions ◽  
Concentration Behavior ◽  
Fractional Choquard Equation

<abstract><p>In this work, we study the existence, multiplicity and concentration behavior of positive solutions for the following problem involving the fractional $ p $-Laplacian</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{eqnarray*} \varepsilon^{ps}(-\Delta )^{s}_{p}u + V(x)|u|^{p-2}u = \varepsilon^{\mu-N}(\frac{1}{|x|^{\mu}}\ast K|u|^{q})K(x)|u|^{q-2}u \hskip0.2cm\text{in}\hskip0.1cm \mathbb{R}^{N}, \end{eqnarray*} $\end{document} </tex-math></disp-formula></p> <p>where $ 0 &lt; s &lt; 1 &lt; p &lt; \infty $, $ N &gt; ps $, $ 0 &lt; \mu &lt; ps $, $ p &lt; q &lt; \frac{p^{*}_{s}}{2}(2-\frac{\mu}{N}) $, $ (-\Delta)^{s}_{p} $ is the fractional $ p $-Laplacian and $ \varepsilon &gt; 0 $ is a small parameter. Under certain conditions on $ V $ and $ K $, we prove the existence of a positive ground state solution and express the location of concentration in terms of the potential functions $ V $ and $ K $. In particular, we relate the number of solutions with the topology of the set where $ V $ attains its global minimum and $ K $ attains its global maximum.</p></abstract>

Modulational instability and multiple rogue wave solutions for the generalized CBS-BK equation

2021 ◽  
pp. 2150408
Author(s):  
Wang Gang ◽  
Jalil Manafian ◽  
Fatma Berna Benli ◽  
Onur Alp İlhan ◽  
Reza Goldaran
Keyword(s):  
Asymptotic Behavior ◽  
Multiple Solutions ◽  
Modulation Instability ◽  
Rogue Wave ◽  
Modulational Instability ◽  
Soliton Solutions ◽  
Hirota’S Bilinear Method ◽  
Bilinear Method ◽  
Wave Solutions ◽  
Multiple Soliton Solutions

An integrable of the generalized Calogero-Bogoyavlenskii-Schiff-Bogoyavlensky-Konopelchenko (CBS-BK) equation is studied, by employing Hirota’s bilinear method the bilinear form is obtained, and the multiple-soliton solutions are constructed. The modified of improved bilinear method has been utilized to investigate multiple solutions. In addition, some graphs including 3D, contour, density, and [Formula: see text]-curves plots of the addressed equation with specific coefficients are shown. Finally, under certain conditions, the asymptotic behavior of the linearization solution is analyzed to prove that the modulation instability is stable for some points.

Existence and Asymptotic Behavior of Positive Solutions for a Class of Quasilinear Schrödinger Equations

2018 ◽  
Vol 18 (1) ◽  
pp. 131-150 ◽  
Author(s):  
Youjun Wang ◽  
Yaotian Shen
Keyword(s):  
Asymptotic Behavior ◽  
Positive Solution ◽  
Positive Solutions ◽  
Ground State ◽  
Schrodinger Equation ◽  
Radial Solution ◽  
Schrödinger Equations ◽  
Quasilinear Schrödinger Equation ◽  
Content Type ◽  
Graphic Content

AbstractIn this paper, we study the quasilinear Schrödinger equation{-\Delta u+V(x)u-\frac{\gamma}{2}(\Delta u^{2})u=|u|^{p-2}u},{x\in\mathbb{R}^{N}}, where{V(x):\mathbb{R}^{N}\to\mathbb{R}}is a given potential,{\gamma>0}, and either{p\in(2,2^{*})},{2^{*}=\frac{2N}{N-2}}for{N\geq 4}or{p\in(2,4)}for{N=3}. If{\gamma\in(0,\gamma_{0})}for some{\gamma_{0}>0}, we establish the existence of a positive solution{u_{\gamma}}satisfying{\max_{x\in\mathbb{R}^{N}}|\gamma^{\mu}u_{\gamma}(x)|\to 0}as{\gamma\to 0^{+}}for any{\mu>\frac{1}{2}}. Particularly, if{V(x)=\lambda>0}, we prove the existence of a positive classical radial solution{u_{\gamma}}and up to a subsequence,{u_{\gamma}\to u_{0}}in{H^{2}(\mathbb{R}^{N})\cap C^{2}(\mathbb{R}^{N})}as{\gamma\to 0^{+}}, where{u_{0}}is the ground state of the problem{-\Delta u+\lambda u=|u|^{p-2}u},{x\in\mathbb{R}^{N}}.

scholarly journals Multiple solutions and ground state solutions for a class of generalized Kadomtsev-Petviashvili equation

2021 ◽  
Vol 19 (1) ◽  
pp. 297-305
Author(s):  
Yuting Zhu ◽  
Chunfang Chen ◽  
Jianhua Chen ◽  
Chenggui Yuan
Keyword(s):  
Bounded Domain ◽  
Ground State ◽  
Multiple Solutions ◽  
Nontrivial Solutions ◽  
Ground State Solutions ◽  
Petviashvili Equation

Abstract In this paper, we study the following generalized Kadomtsev-Petviashvili equation u t + u x x x + ( h ( u ) ) x = D x − 1 Δ y u , {u}_{t}+{u}_{xxx}+{\left(h\left(u))}_{x}={D}_{x}^{-1}{\Delta }_{y}u, where ( t , x , y ) ∈ R + × R × R N − 1 \left(t,x,y)\in {{\mathbb{R}}}^{+}\times {\mathbb{R}}\times {{\mathbb{R}}}^{N-1} , N ≥ 2 N\ge 2 , D x − 1 f ( x , y ) = ∫ − ∞ x f ( s , y ) d s {D}_{x}^{-1}f\left(x,y)={\int }_{-\infty }^{x}f\left(s,y){\rm{d}}s , f t = ∂ f ∂ t {f}_{t}=\frac{\partial f}{\partial t} , f x = ∂ f ∂ x {f}_{x}=\frac{\partial f}{\partial x} and Δ y = ∑ i = 1 N − 1 ∂ 2 ∂ y i 2 {\Delta }_{y}={\sum }_{i=1}^{N-1}\frac{{\partial }^{2}}{{\partial }_{{y}_{i}}^{2}} . We get the existence of infinitely many nontrivial solutions under certain assumptions in bounded domain without Ambrosetti-Rabinowitz condition. Moreover, by using the method developed by Jeanjean [13], we establish the existence of ground state solutions in R N {{\mathbb{R}}}^{N} .

Existence of a Ground State Solution for a Quasilinear Schrödinger Equation

2014 ◽  
Vol 14 (4) ◽  
Author(s):  
Xiang-dong Fang ◽  
Zhi-qing Han
Keyword(s):  
Schrödinger Equation ◽  
Ground State ◽  
Schrodinger Equation ◽  
Ground State Solution ◽  
State Solution ◽  
Quasilinear Schrödinger Equation

AbstractIn this paper we are concerned with the quasilinear Schrödinger equation−Δu + V(x)u − Δ(uwhere N ≥ 3, 4 < p < 4N/(N − 2), and V(x) and q(x) go to some positive limits V

DFT and TD-DFT study of the enol and thiol tautomers of 3-thioxopropanal in the ground and first singlet excited states

2017 ◽  
Vol 16 (04) ◽  
pp. 1750034 ◽  
Author(s):  
Kolsoom Shayan ◽  
Alireza Nowroozi
Keyword(s):  
Excited States ◽  
Ground State ◽  
Global Minimum ◽  
Substitution Effects ◽  
Structure Stability ◽  
Stable Form ◽  
Orbital Parameters ◽  
Td Dft ◽  
Intramolecular Hydrogen ◽  
Linear Correlations

In the first part of this paper, a comprehensive theoretical study of molecular structure, stability, intramolecular hydrogen bond (IMHB) and [Formula: see text]-electron delocalization ([Formula: see text]-ED) of the enol and thiol tautomers of 3-thioxopropanal (TPA) in the ground state is performed. In this regard, all of the plausible conformations of TPA at M06-2X/6-311[Formula: see text]G(d,p) are optimized and a variety of theoretical levels are employed to identify the global minimum. Our calculations show that E1 is the most stable form that is in contrast to the results of Gonzalez et al. [J Phys Chem 101: 9710, 1997]. In order to elucidate this duality, the IMHB and [Formula: see text]-ED of chelated forms (E1 and T1) have been extensively investigated. So, it is found that both of the IMHB analysis and [Formula: see text]-ED concepts emphasize on the E1, as the global minimum. In the second part of this study, a set of simple electron-withdrawing and electron-donating substituents such as CN, F, Cl, CH3 and NH2 have been considered to evaluate their effects on the IMHB of the first singlet excited state of E1 and T1 at TD-DFT/6–311[Formula: see text]G(d,p) level of theory. According to our analysis, it was found that the IMHB strength of the excited states are much weaker than the ground states. Surprisingly, the IMHB of thiol derivatives is stronger than the enol ones in contrast to the ground state. Furthermore, the substitution effects in the ground and excited states are significantly different. Finally, various linear correlations between the IMHB energies with geometrical, topological and molecular orbital parameters are obtained.

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