Positive and Nodal Solutions For a Nonlinear Schrödinger Equation with Indefinite Potential

Furtado, Marcelo F.; Maia, Liliane A.; Medeiros, Everaldo S.

doi:10.1515/ans-2008-0207

Cited by 63 publications

(35 citation statements)

References 26 publications

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“…It is interesting to note that in the last decades, the existence and multiplicity of positive and nodal solutions of classical elliptic problems have been widely investigated; see previous studies [21][22][23][24][25][26][27][28] and references therein. Specially, some results on nodal solutions of nonlinear elliptic equations involving different operators have been obtained by combining minimax method with invariant sets of descending flow, such as Laplacian operator, 23,25,26 p−Laplacian operator, 24 and Schrödinger operator.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Nodal solutions for fractional elliptic equations involving exponential critical growth

Souza

Severo

Rêgo

2020

Math Methods in App Sciences

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In the present paper, we study the existence of least energy nodal solution for a Dirichlet problem driven by the 12−Laplacian operator of the following type: false(−normalΔfalse)12u+Vfalse(xfalse)u=ffalse(ufalse)0.3em0.3emin0.3emfalse(a,bfalse),u=00.3em0.3emin.5emdouble-struckR∖false(a,bfalse), where V:false[a,bfalse]→false[0,+∞false) is a continuous potential and ffalse(tfalse) is a nonlinearity that grows like expfalse(t2false) as t→+∞. By using the constraint variational method and quantitative deformation lemma, we obtain a least energy nodal solution u for the given problem. Moreover, we show that the energy of u is strictly larger than twice the ground state energy.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Nodal solutions for fractional elliptic equations involving exponential critical growth

Souza

Severo

Rêgo

2020

Math Methods in App Sciences

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“…Under similar assumptions on f , Furtado, Maia and Medeiros in [13] obtained nodal solutions for (1.5) by seeking minimizers on the sign-changing Nehari manifold when V may change sign and satisfy mild integral conditions. Nodal solutions of (1.5) in bounded domains are proved to exist by Noussair and Wei in [30] via the Ekeland variational principle and the implicit function theorem, and by Bartsch and Weth in [6] based on the variational method together with the Brouwer degree theory.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

The existence of least energy nodal solutions for some class of Kirchhoff equations and Choquard equations inRN

2015

Journal of Mathematical Analysis and Applications

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In this paper, we study the existence of least energy nodal solutions for a class of Kirchhoff type problems in R N . Since the Kirchhoff equation is a nonlocal one, the variational setting to look for sign-changing solutions is different from the local cases. By using constrained minimization on the sign-changing Nehari manifold, we prove the Kirchhoff problem has a least energy nodal solution with its energy exceeding twice the least energy. As a co-product of our approaches, we obtain the existence of least energy sign-changing solution for Choquard equations in R N and show that the sign-changing solution has an energy strictly larger than the least energy and less than twice the least energy.

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“…By elliptic estimates,

u_{0} \in L^{\infty} false(R^{4} false)

. Hence, it follows from Lemma 8.9 of that

\int_{R^{4}} (| u_{n} {false|}^{2} u_{n} - {false| u_{0} false|}^{2} u_{0}) v_{n}^{1} d x = \int_{R^{4}} {(||, v_{n}^{1})}^{4} d x + o false(1 false) .

Hence, it follows from the Brezis–Lieb lemma that

o false(1 false) = {(‖‖, v_{n}^{1})}^{2} + b (| \nabla u_{0} {false|}_{2}^{2} {|\nabla v_{n}^{1}|}_{2}^{2} + {|\nabla v_{n}^{1}|}_{2}^{4}) - δ β {(||, v_{n}^{1})}_{4}^{4} .

On the other hand, by the Brezis–Lieb lemma, we have

false∥ u_{n} ∥^{2} = {(‖‖, v_{n}^{1})}^{2} + false∥ u_{0} ∥^{2} + o (1), false| u_{n} |_{4}^{4} = {(||, v_{n}^{1})}_{4}^{4} + {| u_{0} |}_{4}^{4} + o false(1 false) .

In view of Lemma 2.4 of , we have

\int_{R^{4}}

…”

Section: Positive Ground Statesmentioning

confidence: 90%

Positive solutions of Kirchhoff type elliptic equations in with critical growth

Liu

Guo

Fang

2016

Mathematische Nachrichten

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In this paper, we study the following Kirchhoff type elliptic problem with critical growth: {−a+b∫double-struckR4false|∇ufalse|2dx▵u+u=ffalse(ufalse)+β|u|2uindouble-struckR4,u∈H1false(R4false),u>0indouble-struckR4,where a,β>0, and b≥0, and the nonlinear growth term |u|2u reaches the Sobolev critical exponent since 2∗=4 for four spatial dimensions. In a non‐radial symmetric function space, we establish a local compactness splitting lemma of critical version to investigate the existence of positive ground state solutions.

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Positive and Nodal Solutions For a Nonlinear Schrödinger Equation with Indefinite Potential

Cited by 63 publications

References 26 publications

Nodal solutions for fractional elliptic equations involving exponential critical growth

Nodal solutions for fractional elliptic equations involving exponential critical growth

The existence of least energy nodal solutions for some class of Kirchhoff equations and Choquard equations inRN

Positive solutions of Kirchhoff type elliptic equations in with critical growth

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