Existence of positive solutions for a class of quasilinear Schrödinger equations with local superlinear nonlinearities

Liang, Zhanping; Gao, Jinfeng; Li, Anran

doi:10.1016/j.jmaa.2019.123732

Cited by 5 publications

(4 citation statements)

References 16 publications

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“…On the other hand, when 𝜅 > 0, the critical power is 𝑝 = 2 * , see [28], where the authors use Pohozaev identity to justify this fact. In the last years, some progress has been made in the case 𝜅 > 0, see [1,3,20,28,36]. Naturally, the above discussion was extended to the case 𝑁 = 2 and 2 * = ∞.…”

Section: Introductionmentioning

confidence: 99%

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Quasilinear Schrödinger equations with unbounded or decaying potentials in dimension 2

Carvalho

Clemente

Albuquerque

2023

Mathematische Nachrichten

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We establish the existence of nontrivial solutions for the following class of quasilinear Schrödinger equations:where 𝜅 is a positive parameter, 𝑉(|𝑥|) and 𝑄(|𝑥|) are continuous functions that can be singular at the origin, unbounded or vanishing at infinity, and the nonlinearity ℎ(𝑠) has critical exponential growth motivated by the Trudinger-Moser inequality. To prove our main result, we apply variational methods together with careful 𝐿 ∞ -estimates.

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Section: Introductionmentioning

confidence: 99%

“…On the other hand, when

\kappa &gt;0

, the critical power is

p=2^{*}

, see [28], where the authors use Pohozaev identity to justify this fact. In the last years, some progress has been made in the case

\kappa &gt;0

, see [1, 3, 20, 28, 36].…”

Section: Introductionmentioning

confidence: 99%

Quasilinear Schrödinger equations with unbounded or decaying potentials in dimension 2

Carvalho

Clemente

Albuquerque

2023

Mathematische Nachrichten

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“…Moreover, [25] obtained least energy nodal solution for (1) when κ > 0 is small enough, λ = 1, K(x) = 1 and µ > 0 is small. Recently, when λ is sufficient large, K(x) = 1 and µ = 0, [13] and [17] obtained the existence of nontrivial solutions for (1) with κ = 2 and κ = 0, respectively. However, up to the best of our knowledge, there is no other papers have studied the existence results of nontrivial solutions for (1) when κ > 0 is large.…”

Section: Introduction Consider the Following Quasilinear Schrödinger Equationsmentioning

confidence: 99%

“…Remark 1. (V, K) ∈ K means that the potentials V (x) and K(x) are vanishing at infinity, which is different from that in [13,17], where the potential function V (x) is assumed to be radially symmetric, bounded and periodic.…”

Section: Introduction Consider the Following Quasilinear Schrödinger Equationsmentioning

confidence: 99%

Positive solutions for critical quasilinear Schrödinger equations with potentials vanishing at infinity

Huang

2022

DCDS-B

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In this paper, we study the existence of positive solutions for the following quasilinear Schrödinger equations<disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} -\triangle u+V(x)u+\frac{\kappa}{2}[\triangle|u|^{2}]u = \lambda K(x)h(u)+\mu|u|^{2^*-2}u, \quad x\in\mathbb{R}^{N}, \end{equation*} $\end{document} </tex-math></disp-formula>where <inline-formula><tex-math id="M1">\begin{document}$ \kappa>0 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M2">\begin{document}$ \lambda>0, \mu>0, h\in C(\mathbb{R}, \mathbb{R}) $\end{document}</tex-math></inline-formula> is superlinear at infinity, the potentials <inline-formula><tex-math id="M3">\begin{document}$ V(x) $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M4">\begin{document}$ K(x) $\end{document}</tex-math></inline-formula> are vanishing at infinity. In order to discuss the existence of solutions we apply minimax techniques together with careful <inline-formula><tex-math id="M5">\begin{document}$ L^{\infty} $\end{document}</tex-math></inline-formula>-estimates. For the subcritical case (<inline-formula><tex-math id="M6">\begin{document}$ \mu = 0 $\end{document}</tex-math></inline-formula>) we can deal with large <inline-formula><tex-math id="M7">\begin{document}$ \kappa>0 $\end{document}</tex-math></inline-formula>. For the critical case we treat that <inline-formula><tex-math id="M8">\begin{document}$ \kappa>0 $\end{document}</tex-math></inline-formula> is small.

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Concentrating Positive Solutions for Quasilinear Schrödinger Equations Involving Steep Potential Well

Yang

Tang

2023

Bull. Malays. Math. Sci. Soc.

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Existence of positive solutions for a class of quasilinear Schrödinger equations with local superlinear nonlinearities

Cited by 5 publications

References 16 publications

Quasilinear Schrödinger equations with unbounded or decaying potentials in dimension 2

Quasilinear Schrödinger equations with unbounded or decaying potentials in dimension 2

Positive solutions for critical quasilinear Schrödinger equations with potentials vanishing at infinity

Concentrating Positive Solutions for Quasilinear Schrödinger Equations Involving Steep Potential Well

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