Multiplicity of Solutions for a Modified Schrödinger-Kirchhoff-Type Equation in  <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant="double-struck">R</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:math>

He, X. T.

doi:10.1155/2015/179540

Discrete Dynamics in Nature and Society

2015

DOI: 10.1155/2015/179540

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Multiplicity of Solutions for a Modified Schrödinger-Kirchhoff-Type Equation in RN

X. T. He

Abstract: We study the existence of infinitely many solutions for a class of modified Schrödinger-Kirchhoff-type equations by the dual method and the nonsmooth critical point theory.

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Existence and multiplicity of nontrivial solutions to the modified Kirchhoff equation without the growth and Ambrosetti–Rabinowitz conditions

Wang

Jia

2021

Electron. J. Qual. Theory Differ. Equ.

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The paper focuses on the modified Kirchhoff equation \begin{align*} -\left(a+b\int_{\mathbb{R}^N}|\nabla u|^2dx\right)\Delta u-u\Delta (u^2)+V(x)u=\lambda f(u), \quad x\in \mathbb{R}^N, \end{align*} where $a,b>0$, $V(x)\in C(\mathbb{R}^N,\mathbb{R})$ and $\lambda<1$ is a positive parameter. We just assume that the nonlinearity $f(t)$ is continuous and superlinear in a neighborhood of $t = 0$ and at infinity. By applying the perturbation method and using the cutoff function, we get existence and multiplicity of nontrivial solutions to the revised equation. Then we use the Moser iteration to obtain existence and multiplicity of nontrivial solutions to the above original Kirchhoff equation. Moreover, the nonlinearity f(t) may be supercritical.

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Existence and multiplicity of nontrivial solutions to the modified Kirchhoff equation without the growth and Ambrosetti–Rabinowitz conditions

Wang

Jia

2021

Electron. J. Qual. Theory Differ. Equ.

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Existence and asymptotic behavior of normalized solutions for the modified Kirchhoff equations in $ \mathbb{R}^3 $

Wang

2022

MATH

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<abstract>This paper is concerned with the following modified Kirchhoff type problem <disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{align*} -\left(a+b\int_{\mathbb{R}^3}|\nabla u|^2\right)\Delta u-u\Delta (u^2)-\lambda u=|u|^{p-2}u, \; \; \; x\in \mathbb{R}^3, \end{align*} $\end{document} </tex-math></disp-formula> where $ a, b > 0 $ are constants and $ \lambda\in \mathbb R $. When $ p=\frac{16}{3} $, we prove that the existence of normalized solution with a prescribed $ L^2 $-norm for the above equation by applying constrained minimization method. Moreover, when $ p\in\left(\frac{10}{3}, \frac{16}{3}\right) $, we prove the existence of mountain pass type normalized solution for the above modified Kirchhoff equation by using the perturbation method. And the asymptotic behavior of normalized solution as $ b\rightarrow 0 $ is analyzed. These conclusions extend some known ones in previous papers.</abstract>

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Multiplicity of Solutions for a Modified Schrödinger-Kirchhoff-Type Equation in RN

Abstract: We study the existence of infinitely many solutions for a class of modified Schrödinger-Kirchhoff-type equations by the dual method and the nonsmooth critical point theory.

Cited by 2 publications

References 21 publications

Existence and multiplicity of nontrivial solutions to the modified Kirchhoff equation without the growth and Ambrosetti–Rabinowitz conditions

Existence and multiplicity of nontrivial solutions to the modified Kirchhoff equation without the growth and Ambrosetti–Rabinowitz conditions

Existence and asymptotic behavior of normalized solutions for the modified Kirchhoff equations in $ \mathbb{R}^3 $

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