Sign-changing solutions of nonlinear elliptic equations

Liu, Zhaoli; Wang, Zhi–Qiang

doi:10.1007/s11464-008-0014-0

Cited by 29 publications

(10 citation statements)

References 62 publications

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“…Over the years, combining minimax methods, the method of invariant sets of descending flow has been a powerful tool in finding sign-changing solutions of elliptic problems. For more progress in this aspect, we refer to [3,4,34,35,45] and the references therein.…”

Section: Resultsmentioning

confidence: 99%

Existence and multiplicity of sign-changing standing waves for a gauged nonlinear Schrödinger equation in $ \newcommand{\R}{\bf {\mathbb R}} \R^2$

2019

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We are concerned with sign-changing solutions of the following gauged nonlinear Schrödinger equation in dimension two including the so-called Chern-Simons termwhere ω, λ > 0, p ∈ (4, 6) andVia a novel perturbation approach and the method of invariant sets of descending flow, we investigate the existence and multiplicity of sign-changing solutions. Moreover, energy doubling is established, i.e., the energy of sign-changing solution w λ is strictly larger than twice that of the ground state energy for λ > 0 large. Finally, for any sequence λ n → ∞ as n → ∞, up to a subsequence, λ

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Section: Resultsmentioning

confidence: 99%

Existence and multiplicity of sign-changing standing waves for a gauged nonlinear Schrödinger equation in $ \newcommand{\R}{\bf {\mathbb R}} \R^2$

2019

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“…Then in the setting of Theorem 4 we obtain one positive, one negative and one sign-changing solutions. If the nonlinearity is odd in u we also obtain infinitely many sign-changing solutions by a standard argument, see [8,9,10,20,23,31].…”

Section: Lemma 6 In (35) We Have (A) H(y)mentioning

confidence: 96%

“…This requires the methods of invariant sets with gradient flows as used in [9](see [23] for more references therein). Let P ± = {u ∈ H | ± u ≥ 0} and…”

Section: Lemma 6 In (35) We Have (A) H(y)mentioning

confidence: 99%

Finding Multiple Solutions to Elliptic PDE with Nonlinear Boundary Conditions

2013

Self Cite

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In this paper, in order to solve an elliptic partial differential equation with a nonlinear boundary condition for multiple solutions, the authors combine a minimax approach with a boundary integral-boundary element method, and identify a subspace and its special expression so that all numerical computation and analysis can be carried out more efficiently based on information of functions only on the boundary. Some mathematical justification of the new approach is established. An efficient and reliable local minimax-boundary element method is developed to numerically search for solutions. Details on implementation of the algorithm are also addressed. The existence and multiplicity of solutions to the problem are established under certain regular assumptions. Some conditions related to convergence of the algorithm and instability of solutions found by the algorithm are verified. To illustrated the method, numerical multiple solutions to some examples on domains with different geometry are displayed with their profile and contour plots.

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“…In addition, the interest in problems of the existence of nodal solutions of nonlinear elliptic equations [9][10][11][12] has recently been growing. This is caused by the fact that nodal solutions arise in applications [12].…”

Section: Introductionmentioning

confidence: 99%

“…This is caused by the fact that nodal solutions arise in applications [12]. Moreover, the interest in the study of nodal solutions is related to a number of well-known unsolved problems of nonlinear analysis, such as determining the number of nodal domains and geometric properties of nodal sets of solutions [13], the problem of the generalization of the Courant theorem on zeros of eigenfunctions to nonlinear problems [14,Chap.…”

Section: Introductionmentioning

confidence: 99%

On the existence of a continuous branch of nodal solutions of elliptic equations with convex-concave nonlinearities

Bobkov

2014

Diff Equat

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We study the existence of nodal solutions of a parametrized family of Dirichlet boundary value problems for elliptic equations with convex-concave nonlinearities. In the main result, we prove the existence of nodal solutions u λ for λ ∈ (−∞, λ * 0 ). The critical value λ * 0 > 0 is found by a spectral analysis procedure according to Pokhozhaev's fibering method. We show that the obtained solutions form a continuous branch (in the sense of level lines of the energy functional) with respect to the parameter λ. Moreover, we prove the existence of an interval (−∞,λ), whereλ > 0, on which this branch consists of solutions with exactly two nodal domains.

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Sign-changing solutions of nonlinear elliptic equations

Abstract: In this survey article, we recall some known results on existence and multiplicity of sign-changing solutions of elliptic equations. Methods for obtaining sign-changing solutions developed in the last two decades will also be briefly revisited.

Cited by 29 publications

References 62 publications

Existence and multiplicity of sign-changing standing waves for a gauged nonlinear Schrödinger equation in $ \newcommand{\R}{\bf {\mathbb R}} \R^2$

Existence and multiplicity of sign-changing standing waves for a gauged nonlinear Schrödinger equation in $ \newcommand{\R}{\bf {\mathbb R}} \R^2$

Finding Multiple Solutions to Elliptic PDE with Nonlinear Boundary Conditions

On the existence of a continuous branch of nodal solutions of elliptic equations with convex-concave nonlinearities

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