A perturbed problem of elliptic system with critical exponent

In this paper, we consider a type of Kirchhoff problem as follows where a, b>0 are given constants, is a small parameter and . We show that if has k critical points near which satisfies some expansion assumption, then by Lyapunov–Schmidt reduction method, we construct multi-peak solutions for small, which concentrate at the k critical points of .

Infinitely many positive multi-bump solutions for fractional Kirchhoff equations

ABSTRACTWe are concerned with the existence of positive nonradial solutions to the following Kirchhoff equation: where and are radial functions having the following expansions: with and . By introducing the Miranda theorem and developing some delicate analysis, we construct infinitely many positive nonradial multibump solutions of this equation under suitable numbers via the Lyapunov–Schmidt reduction method, whose maximum points lie on the top and bottom circles of a cylinder close to infinity. These nonradial multibump solutions are different from the ones obtained in a previous study. This result complements and extends the previous results in the literature.

This paper is concerned with the existence of positive solutions of the nonlinear elliptic problem $-\Delta u+ a(x)u=u^{(N+2)/(N-2)}$, $a(x)\ge 0$, with Neumann boundary conditions in a half-space $\Pi \, \s \, {\Bbb R}^N $, $N \ge 3$. The main feature of the problem is a double lack of compactness due to the unboundedness of the domain and the presence of the critical Sobolev exponent. The solutions are searched using variational methods, although the functional related to the problem does not satisfy the Palais--Smale compactness condition. We observe that the problem considered has no solutions if $a(x)$ is a positive constant; conditions on $a(x)$ are given sufficient to guarantee existence and multiplicity of positive solutions.

In this paper, we study the following critical system with fractional Laplacian: \t\t\t{(−Δ)su+λ1u=μ1|u|2∗−2u+αγ2∗|u|α−2u|v|βin Ω,(−Δ)sv+λ2v=μ2|v|2∗−2v+βγ2∗|u|α|v|β−2vin Ω,u=v=0in RN∖Ω,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\textstyle\\begin{cases} (-\\Delta)^{s}u+\\lambda_{1}u=\\mu_{1}|u|^{2^{\\ast}-2}u+\\frac{\\alpha \\gamma}{2^{\\ast}}|u|^{\\alpha-2}u|v|^{\\beta} & \\text{in } \\Omega, \\\\ (-\\Delta)^{s}v+\\lambda_{2}v= \\mu_{2}|v|^{2^{\\ast}-2}v+\\frac{\\beta \\gamma}{2^{\\ast}}|u|^{\\alpha}|v|^{\\beta-2}v & \\text{in } \\Omega, \\\\ u=v=0 & \\text{in } \\mathbb{R}^{N}\\setminus\\Omega, \\end{cases} $$\\end{document} where (-Delta)^{s} is the fractional Laplacian, 0< s<1, mu_{1},mu_{2}>0, 2^{ast}=frac{2N}{N-2s} is a fractional critical Sobolev exponent, N>2s, 1<alpha, beta<2, alpha+beta=2^{ast}, Ω is an open bounded set of mathbb{R}^{N} with Lipschitz boundary and lambda_{1},lambda_{2}>-lambda_{1,s}(Omega), lambda_{1,s}(Omega) is the first eigenvalue of the non-local operator (-Delta)^{s} with homogeneous Dirichlet boundary datum. By using the Nehari manifold, we prove the existence of a positive ground state solution of the system for all gamma>0. Via a perturbation argument and using the topological degree and a pseudo-gradient vector field, we show that this system has a positive higher energy solution. Then the asymptotic behaviors of the positive ground state solutions are analyzed when gammarightarrow0.

Existence and non-existence of solutions for a class of Monge–Ampère equations

In this paper, we investigate a predator–prey system with nonlinear prey-taxis under Neumann boundary condition. For a class of chemotactic sensitive functions, we obtain the existence and boundedness of global classical solutions for initial boundary value problems in arbitrary dimensional space. In addition, we also study the local stability of the constant steady state solution, and obtain the global asymptotic stability of the steady state solution under different predation intensity by constructing appropriate Lyapunov functions. Furthermore, the steady state bifurcation, Hopf bifurcation and fold-Hopf Singularity are analysed in detail by using Lyapunov–Schmidt reduction method.

Lyapunov–Schmidt reduction algorithm for three-dimensional discrete vortices

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).

Periodic solutions of a nonlinear suspension bridge equation with damping and nonconstant load

A strongly coupled elliptic system under the homogeneous Dirichlet boundary condition denoting the steady-state system of the Lotka-Volterra two-species competitive system with cross-diffusion effects is considered. By using the implicit function theorem and the Lyapunov-Schmidt reduction method, the existence of the positive solutions bifurcating from the trivial solution is obtained. Furthermore, the stability of the bifurcating positive solutions is also investigated by analyzing the associated characteristic equation.

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 $$\cup _{j\in J}a_j$$ .

In this paper, a system of elliptic equations is investigated, which involves multiple critical Sobolev exponents and singular points. The best Sobelev constant related to the system is studied, which is verified to be independent of the location of singular points. By a variant of the concentration compactness principle and the mountain-pass argument, the existence of positive solutions to the system is proved. At last, the existence of sign-changing solutions to the system is also established on the basis of the mountain-pass-type positive solutions.

An elliptic system subject to the homogeneous Dirichlet boundary con- dition denoting the steady-state system of a two-species predator-prey reaction– diffusion system with the modified Leslie–Gower and Holling-type II schemes is con- sidered. By using the Lyapunov–Schmidt reduction method, the bifurcation of the positive solution from the trivial solution is demonstrated and the approximated ex- pressions of the positive solutions around the bifurcation point are also given accord- ing to the implicit function theorem. Finally, by applying the linearized method, the stability of the bifurcating positive solution is also investigated. The results obtained in the present paper improved the existing ones.

In this article, we study the existence of non-radial positive solutions of the Schrodinger-Poisson system $$\displaylines{ -\Delta u+u+V(|x|)\Phi(x)u =Q(|x|) |u|^{p-1}u, \quad x\in \mathbb{R}^3, \cr -\Delta\Phi=V(|x|)u^2,\quad x\in \mathbb{R}^3, }$$ where \(1 &lt; p&lt; 5\) and $V, Q$ are radial potential functions. By developing some refined estimates, via the Lyapunov-Schmidt reduction method, we construct infinitely many multi-bump solutions when \(V, Q\) have some suitable algebraical decay at infinity. The maximum points of those multi-bump solutions are located on the top and bottom circles of a cylinder. This result not only gives a new type of multi-bump solutions but also extends the existence of multi-bump solutions to a general class of potential functions with a relatively slow decay rate at infinity. For more information see https://ejde.math.txstate.edu/Volumes/2025/72/abstr.html