Lyapunov Schmidt Reduction Method Research Articles

A Landesman–Lazer Local Condition for Semilinear Elliptic Problems

The objective of this paper is to study the existence, multiplicity and non existence of solutions for semilinear elliptic problems under a local Landesman–Lazer condition. There is no growth restriction at infinity on the nonlinear term and it may change sign. In order to establish the existence of solution we combine the Lyapunov–Schmidt reduction method with truncation and approximation arguments via bootstrap methods. In our applications we also consider the existence of a bifurcation point which may have multiple positive solutions for a fixed value of the parameters.

A gluing approach for the fractional Yamabe problem with isolated singularities

Abstract We construct solutions for the fractional Yamabe problem that are singular at a prescribed number of isolated points. This seems to be the first time that a gluing method is successfully applied to a non-local problem in order to construct singular solutions. There are two main steps in the proof: to construct an approximate solution by gluing half bubble towers at each singular point, and then an infinite-dimensional Lyapunov–Schmidt reduction method, that reduces the problem to an (infinite-dimensional) Toda-type system. The main technical part is the estimate of the interactions between different bubbles in the bubble towers.

Stability and bifurcation with singularity for a glycolysis model under no-flux boundary condition

In this paper, a glycolysis model subject to no-flux boundary condition is considered. First, by discussing the corresponding characteristic equation, the stability of constant steady state solution is discussed, and the Turing's instability is shown. Next, based on Lyapunov-Schmidt reduction method and singularity theory, the multiple stationary bifurcations with singularity are analyzed. In particular, under no-flux boundary condition we show the existence of nonconstant steady state solution bifurcating from a double zero eigenvalue, which is always excluded in most existing works. Also, the stability, bifurcation direction and multiplicity of the bifurcation steady state solutions are investigated by the singularity theory. Finally, the theoretical results are confirmed by numerical simulations. It is also shown that there is no Hopf bifurcation on basis of the condition \begin{document}$ (C) $\end{document} .

Vortex structures for some geometric flows from pseudo-Euclidean spaces

For some geometric flows (such as wave map equations, Schrödinger flows) from pseudo-Euclidean spaces to a unit sphere contained in a three-dimensional Euclidean space, we construct solutions with various vortex structures (vortex pairs, elliptic/hyperbolic vortex circles, and also elliptic vortex helices). The approaches base on the transformations associated with the symmetries of the nonlinear problems, which will lead to two-dimensional elliptic problems with resolution theory given by the finite-dimensional Lyapunov-Schmidt reduction method in nonlinear analysis.

Multi-peak positive solutions to a class of Kirchhoff equations

In the present paper, we consider the nonlocal Kirchhoff problem$$-\left(\epsilon^2a+\epsilon b\int_{{\open R}^{3}}\vert \nabla u \vert^{2}\right)\Delta u+V(x)u=u^{p}, \quad u \gt 0 \quad {\rm in} {\open R}^{3},$$ where a, b>0, 1<p<5 are constants, ϵ>0 is a parameter. Under some mild assumptions on the function V, we obtain multi-peak solutions for ϵ sufficiently small by Lyapunov–Schmidt reduction method. Even though many results on single peak solutions to singularly perturbed Kirchhoff problems have been derived in the literature by various methods, there exist no results on multi-peak solutions before this paper, due to some difficulties caused by the nonlocal term $\left(\int_{{\open R}^{3}} \vert \nabla u \vert^{2}\right)\Delta u$. A remarkable new feature of this problem is that the corresponding unperturbed problem turns out to be a system of partial differential equations, but not a single Kirchhoff equation, which is quite different from most of the elliptic singular perturbation problems.

Embedded tori with prescribed mean curvature

We construct a sequence of compact, oriented, embedded, two-dimensional surfaces of genus one into Euclidean 3-space with prescribed, almost constant, mean curvature of the form H(X)=1+A|X|−γ for |X| large, when A<0 and γ∈(0,2). Such surfaces are close to sections of unduloids with small neck-size, folded along circumferences centered at the origin and with larger and larger radii. The construction involves a deep study of the corresponding Jacobi operators, an application of the Lyapunov–Schmidt reduction method and some variational argument.

Non-degeneracy of positive solutions of Kirchhoff equations and its application

A non-degeneracy theorem for positive solutions of the standard Kirchhoff model is proved in all cases except for n ≥ 5 and b ∫ R n | ∇ Q | 2 d x = 2 ( n − 4 ) n − 4 2 / ( n − 2 ) n − 2 2 . In particular, for dimensions n ≥ 5 and b ∫ R n | ∇ Q | 2 d x < 2 ( n − 4 ) n − 4 2 / ( n − 2 ) n − 2 2 , we show that there exist two non-degenerate positive solutions of the Kirchhoff equations, which seem to be completely different from the result of the standard Schrödinger equation. The effects of the non-local term on the positive solution set are also studied. We show that the non-local term has no effect on the structure of the positive solution set for dimensions 1 ≤ n ≤ 3 and the effects eventually appear for dimensions n ≥ 4 . With the non-degeneracy property of positive solutions of the limit problem, we construct concentrated solutions of a singularly perturbed Kirchhoff problem via the well-known Lyapunov–Schmidt reduction method. For dimensions n ≥ 5 , the existence result of concentrated solutions is completely new and is not implied by the main result of G.M. Figueiredo et al. in [19] , where a generalized singularly perturbed Kirchhoff problem was considered.

Conformal scalar curvature equation on Sn: Functions with two close critical points (Twin Pseudo-Peaks)

By using the Lyapunov–Schmidt reduction method without perturbation, we consider existence results for the conformal scalar curvature on [Formula: see text] ([Formula: see text]) when the prescribed function (after being projected to [Formula: see text]) has two close critical points, which have the same value (positive), equal “flatness” (“twin”; flatness [Formula: see text]), and exhibit maximal behavior in certain directions (“pseudo-peaks”). The proof relies on a balance between the two main contributions to the reduced functional — one from the critical points and the other from the interaction of the two bubbles.

Bifurcation of Solution in Singularly Perturbed ODEs by Using Lyapunov Schmidt Reduction

This paper aims to study the bifurcation of solution in singularly perturbed ODEs: the hypothesis the bifurcation of solution in the ODE system will be studied by effect of the system by using Lyapunov Schmidt reduction. Is the study of behaviour of solution of singularly perturbed ODEs when perturbation parameter The bifurcation of solution in this kind of ordinary differential equation was studied in n-dimensional. Sufficient conditions for the system to undergoes (fold,transcritical and pitchfork) bifurcation are given. The ODE will be reduced to an equivalent system by using Lyapunov Schmidt reduction method. Moreover, for this purpose of obtaining curve of the system (Fast-Slow system).

Pairs of solutions for Robin problems with an indefinite and unbounded potential, resonant at zero and infinity

We consider a semilinear Robin problem driven by the Laplacian plus an indefinite and unbounded potential and a Caratheodory reaction term which is resonant both at zero and $$\pm \infty $$ . Using the Lyapunov–Schmidt reduction method and critical groups (Morse theory), we show that the problem has at least two nontrivial smooth solutions.

Multi-scale reduced order models for transient simulation of multi-layered monolith reactors

We develop and use a multi-scale averaging technique based on the Lyapunov-Schmidt (L-S) reduction method to obtain reduced order models for monolith reactors with multiple washcoat layers. These models are expressed in terms of phase-averaged concentration modes and inter as well as intra-phase (dispersion) fluxes. The various fluxes are determined by the local equations that are linear in the concentration modes and fluxes. It is shown that the structure of the multi-phase averaged models derived by the L-S method is fundamentally different from the traditional intuitively written models. We provide physical interpretation of the various inter and intra-phase transfer coefficients appearing in the reduced order models and also expressions for them for common geometries. The accuracy of the reduced order models is illustrated by comparing with the exact solutions for selected cases and also with other models in the literature. The usefulness of the reduced order models is illustrated by two examples of dual layered monolith reactors used in diesel oxidation catalysts and hydrocarbon traps.

Effect of the protection zone on coexistence of the species for a ratio-dependent predator-prey model

This paper deals with the effect of the protection zone Ω0 on coexistence of the species for a ratio-dependent predator-prey model. We obtain a critical value λ1N(bδ(x),Ω) which is less than the well-known critical value λ1D(Ω0) obtained in the previous literatures. Furthermore, we show that if λ>λ1N(bδ(x),Ω), then the prey persists regardless of the growth rates of the predator; while if λ≤λ1N(bδ(x),Ω), then there exists a real number μ⁎, such that the prey is ultimately extinct when μ>μ⁎. As for λ<λ1N(bδ(x),Ω) and μ<μ⁎, the curve of the positive steady state solutions of the model emanating from (λ,0;−c) ends at a singular point (0,0;μ2). Meantime, by using the Lyapunov–Schmidt reduction method, we obtain a fine profile of its bifurcation diagrams, and the uniqueness or multiplicity of its positive steady state solutions. In addition, as generally expected, the chances of survival of the prey will increase with the size of the protection zone.

Segregated vector solutions with multi-scale spikes for nonlinear coupled elliptic systems

In this paper, we consider the following nonlinear coupled elliptic system(Aε){−ε2Δu+P(x)u=μ1u3+βuv2in RN,−ε2Δv+Q(x)v=μ2v3+βu2vin RN,u>0,v>0in RN,u→0,v→0as |x|→+∞, where ε>0 is a parameter, μ1,μ2>0 and β>0 are constants, and P(x) and Q(x) are two nonnegative, smooth functions with different nondegenerate critical points and separated zero sets. Due to the Lyapunov–Schmidt reduction method and the Maximum Principle, we show that when β is less than a small positive number, there exists an ε0>0 such that for any 0<ε<ε0, (Aε) has a segregated vector solution where (uε,vε) and uε is trapped in a neighborhood of the nondegenerate critical points of P(x) as well as the zero sets of P(x), and vε is trapped in a neighborhood of the nondegenerate critical points of Q(x) as well as the zero sets of Q(x). Moreover, the amplitudes of uε (res vε) around the nondegenerate critical points and the zero sets of P(x) (res Q(x)) are of a different order compared with ε. To the best of our knowledge, these multi-scale solutions to the system have not been obtained previously.

Existence of the solutions of a reaction cross-diffusion model for two species

In this paper, we investigate a two-cooperative species model with reaction cross-diffusion under the Dirichlet boundary value condition. By applying Lyapunov–Schmidt reduction method and some important formulas, we obtain the existence of coexistence solutions under some given conditions.

Multi-bump bound states for a Schrödinger system via Lyapunov–Schmidt Reduction

Motivated by problems arising in nonlinear optics and Bose–Einstein condensates, we consider in $$\mathbb R^N$$ ( $$N \le 3$$ ) the following $$n \times n$$ system of coupled Schrodinger equations where $$\varepsilon >0$$ is a parameter, $$\beta _{ij}$$ are constants satisfying $$\beta _{ii} > 0$$ , and $$V_i$$ are positive potentials that admit some common critical points $$a_1, \ldots ,a_k$$ satisfying certain non-degenerate assumption. Then for any subsets $$J\subset \{1,2,\ldots ,k\}$$ , using a Lyapunov–Schmidt reduction method, we prove the existence of multi-bump bound solutions which as $$\varepsilon \rightarrow 0$$ concentrate on $$\cup _{j\in J}a_j$$ .

On a free boundary problem and minimal surfaces

From minimal surfaces such as Simons' cone and catenoids, using refined Lyapunov–Schmidt reduction method, we construct new solutions for a free boundary problem whose free boundary has two components. In dimension 8, using variational arguments, we also obtain solutions which are global minimizers of the corresponding energy functional. This shows that the theorem of Valdinoci et al. [41,42] is optimal.

Existence, stability, and dynamics of harmonically trapped one-dimensional multi-component solitary waves: The near-linear limit

In the present work, we consider a variety of two-component, one-dimensional states in nonlinear Schrödinger equations in the presence of a parabolic trap, inspired by the atomic physics context of Bose-Einstein condensates. The use of Lyapunov-Schmidt reduction methods allows us to identify persistence criteria for the different families of solutions which we classify as (m, n), in accordance with the number of zeros in each component. Upon developing the existence theory, we turn to a stability analysis of the different configurations, using the Krein signature and the Hamiltonian-Krein index as topological tools identifying the number of potentially unstable eigendirections for each branch. A perturbation expansion for the eigenvalue problems associated with nonlinear states found near the linear limit permits us to obtain explicit asymptotic expressions for the eigenvalues. Finally, when the states are found to be unstable, typically by virtue of Hamiltonian Hopf bifurcations, their dynamics is studied in order to identify the nature of the respective instability. The dynamics is generally found to lead to a vibrational evolution over long time scales.

Number of synchronized and segregated interior spike solutions for nonlinear coupled elliptic systems

In this paper, we consider the following nonlinear coupled elliptic systems(Aε){−ε2Δu+u=μ1u3+βuv2in Ω,−ε2Δv+v=μ2v3+βu2vin Ω,u>0,v>0in Ω,∂u∂ν=∂v∂ν=0on ∂Ω, where ε>0, μ1>0, μ2>0, β∈R, and Ω is a bounded domain with smooth boundary in R3. Due to Lyapunov–Schmidt reduction method, we proved that (Aε) has at least O(1ε3|ln⁡ε|) synchronized and segregated vector solutions for ε small enough and some β∈R. Moreover, for each m∈(0,3) there exist synchronized and segregated vector solutions for (Aε) with energies in the order of ε3−m. Our result extends the result of Lin, Ni and Wei [20], from the Lin–Ni–Takagi problem to the nonlinear elliptic systems.

Construction of blow-up sequences for the prescribed scalar curvature equation on $$S^n$$ S n . III. Aggregated and towering blow-ups

Using annular domains introduced in Part II (Leung, Calc Var PDE 46:1–29, 2013), we present constructive results on blow-up sequences of infinite number of solutions for the (prescribed and fixed) conformal scalar curvature equation on \(S^n\) (\(n\ge 6\)), including aggregated and towering blow-ups. The constructions make use of the Lyapunov–Schmidt reduction method, and count on the hyperbolic structure on the collection of standard bubbles.

Bifurcations and stability of standing waves in the nonlinear Schrödinger equation on the tadpole graph

We develop a detailed analysis of edge bifurcations of standing waves in the nonlinear Schrödinger (NLS) equation on a tadpole graph (a ring attached to a semi-infinite line subject to the Kirchhoff boundary conditions at the junction). It is shown in the recent work [] by using explicit Jacobi elliptic functions that the cubic NLS equation on a tadpole graph admits a rich structure of standing waves. Among these, there are different branches of localized waves bifurcating from the edge of the essential spectrum of an associated Schrödinger operator.We show by using a modified Lyapunov-Schmidt reduction method that the bifurcation of localized standing waves occurs for every positive power nonlinearity. We distinguish a primary branch of never vanishing standing waves bifurcating from the trivial solution and an infinite sequence of higher branches with oscillating behavior in the ring. The higher branches bifurcate from the branches of degenerate standing waves with vanishing tail outside the ring.Moreover, we analyze stability of bifurcating standing waves. Namely, we show that the primary branch is composed by orbitally stable standing waves for subcritical power nonlinearities, while all nontrivial higher branches are linearly unstable near the bifurcation point. The stability character of the degenerate branches remains inconclusive at the analytical level, whereas heuristic arguments based on analysis of embedded eigenvalues of negative Krein signatures support the conjecture of their linear instability at least near the bifurcation point. Numerical results for the cubic NLS equation show that this conjecture is valid and that the degenerate branches become spectrally stable far away from the bifurcation point.