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In this paper, we are interested in the following nonlinear elliptic equation (0.1) { Δ u − V (x) u + u p = 0 in R N , u ∈ H 1 (R N) , where N ≥ 1 , p is subcritical, and V (x) is continuous, oscillates and decays to a positive constant at infinity. We use the localized energy technique and the Lyapunov–Schmidt reduction method to prove that the problem (0.1) has infinitely many positive solutions. Our result seems to be new for the existence of infinitely many positive solutions for elliptic equations with oscillatory potentials. [ABSTRACT FROM AUTHOR]

We consider the following anisotropic sinh-Poisson type equation with a Hardy or Hénon term: 0.1 { − div (a (x) ∇ u) + a (x) u = ε 2 a (x) | x − q | 2 α (e u − e − u) in Ω , ∂ u ∂ n = 0 , on Ω , where ε > 0 , q ∈ Ω ¯ ⊂ R 2 , α ∈ (− 1 , ∞) ∖ N , Ω ⊂ R 2 is a smooth bounded domain, n is the unit outward normal vector of ∂Ω, and anisotropic coefficient a (x) is a smooth positive function defined on Ω̄. From the finite-dimensional reduction method, we proved that the problem (0.1) has a sequence of sign-changing solutions with arbitrarily many interior spikes accumulating to q, provided q ∈ Ω is a strict local maximizer of a (x) . However, if q ∈ ∂ Ω is a strict local maximum point of a (x) and satisfies 〈 ∇ a (q) , n 〉 = 0 , we proved that (0.1) has a family of sign-changing solutions with arbitrarily many mixed interior and boundary spikes accumulating to q. Under the same condition, we could also construct a sequence of blow-up solutions to the following problem { − div (a (x) ∇ u) + a (x) u = ε 2 a (x) | x − q | 2 α e u in Ω , ∂ u ∂ n = 0 , on ∂ Ω. [ABSTRACT FROM AUTHOR]

Abstract We consider the following anisotropic sinh-Poisson type equation with a Hardy or Hénon term: 0.1 { − div ( a ( x ) ∇ u ) + a ( x ) u = ε 2 a ( x ) | x − q | 2 α ( e u − e − u ) in Ω , ∂ u ∂ n = 0 , on Ω , $$ \left \{ \textstyle\begin{array}{l@{\quad}l} -\mathrm{div}(a(x)\nabla u)+ a(x)u=\varepsilon ^{2}a(x)|x-q|^{2\alpha}(e^{u}-e^{-u}) &\text{in $\Omega $,} \\ \frac{\partial u}{\partial n}=0, &\text{on $\Omega $,} \end{array}\displaystyle \right . $$ where ε > 0 $\varepsilon >0$ , q ∈ Ω ¯ ⊂ R 2 $q\in \bar{\Omega}\subset \mathbb{R}^{2}$ , α ∈ ( − 1 , ∞ ) ∖ N $\alpha \in (-1,\infty )\backslash \mathbb{N}$ , Ω ⊂ R 2 $\Omega \subset \mathbb{R}^{2}$ is a smooth bounded domain, n is the unit outward normal vector of ∂Ω, and anisotropic coefficient a ( x ) $a(x)$ is a smooth positive function defined on Ω̄. From the finite-dimensional reduction method, we proved that the problem (0.1) has a sequence of sign-changing solutions with arbitrarily many interior spikes accumulating to q, provided q ∈ Ω $q\in \Omega $ is a strict local maximizer of a ( x ) $a(x)$ . However, if q ∈ ∂ Ω $q\in \partial \Omega $ is a strict local maximum point of a ( x ) $a(x)$ and satisfies 〈 ∇ a ( q ) , n 〉 = 0 $\langle \nabla a(q),n \rangle =0$ , we proved that (0.1) has a family of sign-changing solutions with arbitrarily many mixed interior and boundary spikes accumulating to q. Under the same condition, we could also construct a sequence of blow-up solutions to the following problem { − div ( a ( x ) ∇ u ) + a ( x ) u = ε 2 a ( x ) | x − q | 2 α e u in Ω , ∂ u ∂ n = 0 , on ∂ Ω . $$ \left \{ \textstyle\begin{array}{l@{\quad}l} -\mathrm{div}(a(x)\nabla u)+ a(x)u=\varepsilon ^{2}a(x)|x-q|^{2\alpha}e^{u} &\text{in $\Omega $,} \\ \frac{\partial u}{\partial n}=0, &\text{on $\partial \Omega $.} \end{array}\displaystyle \right . $$

In this paper, we study the existence of solutions for the following nonlocal superlinear elliptic problem (0.1) { (− Δ) s u = β (x) u p − ε in Ω , u = 0 in R n ﹨ Ω , where s ∈ (0 , 1) , n > 2 s , p : = n + 2 s n − 2 s is the Sobolev critical exponent, Ω ⊂ R n is a smooth bounded domain with Lipschitz boundary, (− Δ) s is the fractional Laplace operator and β ∈ C 2 (Ω ‾) is a bounded positive continuous function. We assume that there exists a nondegenerate critical point ξ ⁎ ∈ ∂ Ω of the restriction of β to the boundary ∂Ω such that ∇ (β (ξ ⁎) − n − 2 s 2 s ) ⋅ η (ξ ⁎) > 0. Given any integer k ≥ 1 , we show that for ε > 0 small enough, problem (0.1) has a positive solution, which is a sum of k bubbles which concentrate at ξ ⁎ as ε tends to zero. Also, we prove the existence of nodal (sign changing) solution whose shape resembles a sum of a positive bubble and a negative bubble near the point ξ ⁎. This work can be seen as a nonlocal analogue of the result by Dávila, Faya and Mahmoudi, see [28]. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the following Schrödinger system { − Δ u + P (x) u = μ 1 u 3 + β 1 u v 2 + β 2 u w 2 , x ∈ R 3 , − Δ v + Q (x) v = μ 2 v 3 + β 1 v u 2 + β 3 v w 2 , x ∈ R 3 , − Δ w + λ w = μ 3 w 3 + β 2 w u 2 + β 3 w v 2 , x ∈ R 3 , where λ > 0 is a positive constant, P (x) , Q (x) are continuous positive radial potentials, μ i (i = 1 , 2 , 3) > 0 and β i (i = 1 , 2 , 3) ∈ R are coupling constants. We mainly investigate the effect of the potentials and the nonlinear coupling on the structure of solutions. Applying the Lyapunov-Schmidt reduction method, we prove the existence of infinitely many positive solutions and sign-changing solutions to the system whose energy can be arbitrarily large. Specifically, we obtain solutions with some of the components synchronized between them while segregated with the rest of the components. Moreover, we also show the existence of another solutions with all components segregated, one of which concentrates at the origin. Our results present vector solutions to the system with different characters. To our knowledge, it is the first time to study systems with three equations involving mixed potentials. [ABSTRACT FROM AUTHOR]

In this paper, we study the following biharmonic elliptic problem involving slightly subcritical non-power nonlinearity {Δ2u = |u|2∗−2u [ln(e+|u|)]휖inΩ,Δu = u = 0 on∂Ω, where Ω ⊂ ℝn is a bounded smooth domain, 2∗ = 2n n−4, n ≥ 5, 휖 > 0 is a small parameter. By Lyapunov–Schmidt procedure, under suitable assumptions on the Robin function related to Ω, we construct the multi-peak solutions which blow-up and concentrate in different points of Ω as 휖 goes to 0. [ABSTRACT FROM AUTHOR]

We study the Dirichlet problem for an elliptic system derived from FitzHugh–Nagummo model as follows: - ε 2 Δ u = f (u) - v , in Ω , - Δ v + γ v = δ ε u , in Ω , u = v = 0 , on ∂ Ω , where Ω represents a bounded smooth domain in R 2 and ε , γ are positive constants. The parameter δ ε > 0 is a constant dependent on ε , and the nonlinear term f(u) is defined as u (u - a) (1 - u) . Here, a is a function in C 2 (Ω) ∩ C 1 (Ω ¯) with its range confined to (0 , 1 2) . Our research focuses on this spatially inhomogeneous scenario whereas the scenario that a is spatially constant has been studied extensively by many other mathematicians. Specifically, in dimension two, we utilize the Lyapunov–Schmidt reduction method to establish the existence of a single interior peak solution. This is contingent upon a mild condition on a, which acts as an indicator of a location-dependent activation threshold for excitable neurons in the biological environment. [ABSTRACT FROM AUTHOR]

In this paper, we study a class of Choquard equation with nonsymmetric potentials: - Δ u + (1 + δ V (x)) u = (| x | - 1 ∗ | u | 2) u in R 3 , where V is a continuous potential and satisfies some nondegeneracy condition. By using localized energy method, we prove that there exists δ 0 > 0 such that for 0 < δ < δ 0 , the Choquard equation has infinitely many solutions with multi-bumps. [ABSTRACT FROM AUTHOR]

In this article, we study the following fractional critical Schrödinger system (- Δ) s u i = μ i u i 3 + β u i ∑ j ≠ i u j 2 + λ i u i in Ω , u i = 0 on R N \ Ω , i = 1 , 2 , … , m , where 0 < s < 1 , μ i > 0 , coupling constant β satisfies either - ∞ < β ≤ β ¯ ( β ¯ > 0 small) or β → - ∞ , 0 < λ i < λ 1 s (Ω) , where λ 1 s (Ω) is the first eigenvalue of (- Δ) s on Ω , with Ω is a smooth bounded domain in R N with N = 4 s . Under some geometric assumptions on Ω , we construct solutions which concentrate and blow up at different points as λ 1 , … , λ m → 0 . [ABSTRACT FROM AUTHOR]

In this paper, we first show the nondegeneracy and asymptotic behavior of ground states for the nonlinear fourth-order Schrödinger equation with mixed dispersion: \[ \delta\Delta^{2}u-\Delta u +u=|u|^{2\sigma}u, \quad u\in H^{2}(\mathbb{R}^{N}), \] δ Δ 2 u − Δu + u = | u | 2 σ u , u ∈ H 2 (R N) , where $ \delta \gt 0 $ δ > 0 is sufficiently small, $ 0 \lt \sigma \lt \frac {2}{(N-2)_+} $ 0 < σ < 2 (N − 2) + , $ \frac {2}{(N-2)_+}=\frac {2}{N-2} $ 2 (N − 2) + = 2 N − 2 for $ N\geq ~3 $ N ≥ 3 and $ \frac {2}{(N-2)_+}=+\infty $ 2 (N − 2) + = + ∞ for N=2,3. This work extends some results in Bonheure, Casteras, Dos Santos, and Nascimento [Orbitally stable standing waves of a mixed dispersion nonlinear Schrödinger equation. SIAM J Math Anal. 2018;50:5027–5071]. Next, suppose $ P(x) $ P (x) and $ Q(x) $ Q (x) are two positive, radial and continuous functions satisfying that as $ r=|x|\rightarrow +\infty $ r = | x | → + ∞ , \[ P(r)=1+\frac{a_1}{r^{m_1}} +O\left(\frac{1}{r^{m_1+\theta_1}}\right),\quad Q(r)=1+\frac{a_2}{r^{m_2}} +O\left(\frac{1}{r^{m_2+\theta_2}}\right), \] P (r) = 1 + a 1 r m 1 + O (1 r m 1 + θ 1 ) , Q (r) = 1 + a 2 r m 2 + O (1 r m 2 + θ 2 ) , where $ a_1, a_2\in \mathbb {R} $ a 1 , a 2 ∈ R , $ m_1, m_2 \gt 1 $ m 1 , m 2 > 1 , $ \theta _1, \theta _2 \gt 0 $ θ 1 , θ 2 > 0. We use the Lyapunov–Schmidt reduction method developed by Wei and Yan [Infinitely many positive solutions for the nonlinear Schrödinger equations in RN. Calc Var. 2010;37:423–439] to construct infinitely many nonradial positive and sign-changing solutions with arbitrary large energy for the following equation: \[ \delta\Delta^{2}u-\Delta u +P(x)u=Q(x)|u|^{2\sigma}u, \quad u\in H^{2}(\mathbb{R}^{N}). \] δ Δ 2 u − Δu + P (x) u = Q (x) | u | 2 σ u , u ∈ H 2 (R N). [ABSTRACT FROM AUTHOR]

In this article, we establish the existence of positive multi-peak solutions to the following elliptic problem - Δ v + (λ + V (x)) v = v p in Ω , v > 0 in Ω , ∫ Ω v 2 d x = ρ , where Ω is a bounded smooth domain of R N or the whole space R N , the exponent p satisfies 1 < p < N + 2 N - 2 for N ≥ 3 and p > 1 for N = 1 , 2 . For the case of mass subcritical, mass critical, and mass supercritical, we shall deal with the effect of ρ on the existence of the solution concentrating at k different points, which belong to either ∂ Ω or Ω , or R N . [ABSTRACT FROM AUTHOR]

In this paper, we consider the following nonlinear Schrödinger–Poisson system in ℝ 3 : 0.1 − Δ u + u + K (| y |) Φ (y) u = Q (| y |) | u | p − 1 u in ℝ 3 , − Δ Φ = K (| y |) u 2 in ℝ 3 , where 1 < p < 5, K(|y|) and Q(|y|) are two radial, positive, and locally Hölder continuous functions satisfying that K (r) = a r m + O 1 r m + 휃 , Q (r) = 1 + b r n + O 1 r n + ε as r → + ∞ , where b ∈ ℝ , a,m,n,휃 and ε are positive constants. Simulated by the work of Wang and Zhao [31], either a > 0, b < 0 or a > 0, b > 0, m > 2n holds, we use the Lyapunov–Schmidt reduction method to construct infinitely many nonradial positive solutions of (0.1) with arbitrary large energy. [ABSTRACT FROM AUTHOR]

We construct multiple solutions to the nonlocal Liouville equation (-Δ)1/2 u = K(x)eu in R. More precisely, for K of the form K(x) = 1 + ε k (x) with ε Є (0, 1) small and k Є C1,α (R) L∞ (R) for some α >0, we prove the existence of multiple solutions to the above equation bifurcating from the bubbles. These solutions provide examples of flat metrics in the half-plane with prescribed geodesic curvature K(x) on its boundary. Furthermore, they imply the existence of multiple ground state soliton solutions for the Calogero--Moser derivative nonlinear Schrödinger equation. [ABSTRACT FROM AUTHOR]

For the incompressible Euler flows in planar bounded domains Ω, we prove that the vortex concentrating at the harmonic center of Ω causes the existence of vortexes with small vorticities near the boundary. These results are very different from the known results for the multivortex flows in which all the local vorticities are of the same orders. [ABSTRACT FROM AUTHOR]

In this paper, we are devoted to studying a Holling type-II predator-prey system with cross diffusion. First of all, by a series of estimates, we establish the global existence of generalized solutions under some proper assumptions. By analyzing the distribution of eigenvalues, we investigate the local and global stability of the constant solution and also consider the steady state bifurcation and Hopf bifurcation near the constant steady state in details. In addition, the nonexistence of non-constant steady states is also investigated when diffusion rate d is large enough. Finally, sufficient conditions ensuring the existence of non-constant steady states are obtained by using Leray-Schauder degree theory. [ABSTRACT FROM AUTHOR]

In this paper, we construct the solutions to the following nonlinear Schrödinger system - ϵ 2 Δ u + P (x) u = μ 1 u p + β u p - 1 2 v p + 1 2 in R N , - ϵ 2 Δ v + Q (x) v = μ 2 v p + β u p + 1 2 v p - 1 2 in R N , where 3 < p < + ∞ , N ∈ { 1 , 2 } , ϵ > 0 is a small parameter, the potentials P, Q satisfy 0 < P 0 ≤ P (x) ≤ P 1 and Q(x) satisfies 0 < Q 0 ≤ Q (x) ≤ Q 1 . We construct the solution for attractive and repulsive cases. When x 0 is a local maximum point of the potentials P and Q and P (x 0) = Q (x 0) , we construct k spikes concentrating near the local maximum point x 0 . When x 0 is a local maximum point of P and x ¯ 0 is a local maximum point of Q, we construct k spikes of u concentrating at the local maximum point x 0 and m spikes of v concentrating at the local maximum point x ¯ 0 when x 0 ≠ x ¯ 0. This paper extends the main results established by Peng and Wang (Arch Ration Mech Anal 208:305–339, 2013) and Peng and Pi (Discrete Contin Dyn Syst 36:2205–2227, 2016), where the authors considered the case N = 3 , p = 3 . [ABSTRACT FROM AUTHOR]

In this paper, we focus on the dynamical properties of a diffusive Lotka–Volterra system with fear effects subjected to the Neuman boundary condition in a bounded domain. The nonexistence of nonconstant solutions is obtained when diffusion rate is sufficiently large. We also establish the existence and multiplicity of the nonhomogeneous steady-state solutions bifurcating from the constant solution of the system by means of the Lyapunov–Schmidt reduction method. In addition, we consider the influence of the intrinsic growth rate on the population dynamics of the model and show that not only can the population density of both predator and prey change by changing the intrinsic growth rate, but also the coexistence equilibrium can possibly be destabilized. At the same time, we also consider the fear effect on the stability, we find that there is no obvious impact on the stability when the fear effect k is small. However, the results on stability will obviously change when the fear effect is large, which are very different from the ODE system (Wang et al. in J Math Biol 73:1179-1204, 2016) without the fear effect. [ABSTRACT FROM AUTHOR]

We consider an autonomous ordinary differential equation that admits a heteroclinic loop. The unperturbed heteroclinic loop consists of two degenerate heteroclinic orbits $ \gamma_1 $ and $ \gamma_2 $. We assume the variational equation along the degenerate heteroclinic orbit $ \gamma_i $ has $ {d_i}\left({{d_i} > 1, i = 1, 2} \right) $ linearly independent bounded solutions. Moreover, the splitting indices of the unperturbed heteroclinic orbits are $ s $ and $ -s $ $ (s\geq 0) $, respectively. In this paper, we study the persistence of the heteroclinic loop under periodic perturbation. Using the method of Lyapunov-Schmidt reduction and exponential dichotomies, we obtained the bifurcation function, which is defined from $ \mathbb{R}^{d_1+d_2+2} $ to $ \mathbb{R}^{d_1+d_2} $. Under some conditions, the perturbed system can have a heteroclinic loop near the unperturbed heteroclinic loop.

We consider the prescribed scalar curvature problem on S N Δ S N v - N (N - 2) 2 v + K ~ (y) v N + 2 N - 2 = 0 on S N , v > 0 in S N , under the assumptions that the scalar curvature K ~ is rotationally symmetric, and has a positive local maximum point between the poles. We prove the existence of infinitely many non-radial positive solutions, whose energy can be made arbitrarily large. These solutions are invariant under some non-trivial sub-group of O(3) obtained doubling the equatorial. We use the finite dimensional Lyapunov–Schmidt reduction method. [ABSTRACT FROM AUTHOR]

We consider the following fractional Schrödinger equation: (0.1) (− Δ) s u + V (y) u = u p , u > 0 in ℝ N , where s ∈ (0, 1), 1 < p < N + 2 s N − 2 s , and V(y) is a positive potential function and satisfies some expansion condition at infinity. Under the Lyapunov-Schmidt reduction framework, we construct two kinds of multi-spike solutions for (0.1). The first k-spike solution uk is concentrated at the vertices of the regular k-polygon in the (y1, y2)-plane with k and the radius large enough. Then we show that uk is non-degenerate in our special symmetric workspace, and glue it with an n-spike solution, whose centers lie in another circle in the (y3, y4)-plane, to construct infinitely many multi-spike solutions of new type. The nonlocal property of (−Δ)s is in sharp contrast to the classical Schrödinger equations. A striking difference is that although the nonlinear exponent in (0.1) is Sobolev-subcritical, the algebraic (not exponential) decay at infinity of the ground states makes the estimates more subtle and difficult to control. Moreover, due to the non-locality of the fractional operator, we cannot establish the local Pohozaev identities for the solution u directly, but we address its corresponding harmonic extension at the same time. Finally, to construct new solutions we need pointwise estimates of new approximation solutions. To this end, we introduce a special weighted norm, and give the proof in quite a different way. [ABSTRACT FROM AUTHOR]

Abstract We are concerned with the question of constructing a new type of solution to the problem with χ ( 2 ) $\chi ^{(2)}$ nonlinearities 0.1 { − Δ u + P ( x ) u = α u v , in R N , − Δ v + Q ( x ) v = α 2 u 2 + β v 2 , in R N , where P ( x ) = P ( | x | ) $P(x)=P(|x|)$ and Q ( x ) = Q ( | x | ) $Q(x)=Q(|x|)$ are positive bounded radial potentials, 3 ≤ N < 6 $3\leq N 0 $\alpha >0$ and α > β $\alpha >\beta $ . Assuming that the potentials P ( x ) $P(x)$ and Q ( x ) $Q(x)$ satisfy certain conditions, the existence of a new type of solutions is proved.

We reduce solution of the Dirichlet problem ($x \in D \subset R^m$) \[ \Delta u(x)+a(x)u(x)=f(x) \quad \mbox{in $D$}, \qquad u=0 \quad \mbox{on $\partial D$} \] to iterative solution of a simpler problem \[ \Delta u=f(x) \; \; \mbox{in $D$}, \; \; u=0 \; \; \mbox{on $\partial D$} \,, \] for which one can use either Fourier series or Green's function method. The method is suitable for numerical computations, particularly when one uses Newton's method for semilinear problems \[ \Delta u+g(x,u)=0 \quad \mbox{in $D$}, \qquad u=0 \quad \mbox{on $\partial D$}, \] in dimensions $m \geq 3$.

In this paper, we investigate a reaction–diffusion model with delay and Robin boundary condition in heterogeneous environment. The existence, multiplicity and stability of spatially nonhomogeneous steady-state solutions and periodic solutions are studied by employing the Lyapunov–Schmidt reduction method. Moreover, the Hopf bifurcation direction is derived. It is observed that Robin boundary condition plays a crucial role in the Hopf bifurcation. More precisely, when the boundary effect is stronger than the interaction of the species within the region, there is no Hopf bifurcation no matter how the time delay τ changes. Finally, we illustrate our general theoretical results by an application to the Nicholson's blowflies model. [ABSTRACT FROM AUTHOR]

In this paper, a diffusive genetic regulatory network under Neumann boundary conditions is considered. First, the criteria for the local stability and diffusion-driven instability of the positive stationary solution without and with diffusion are investigated, respectively. Moreover, Turing regions and pattern formation are obtained in the plane of diffusion coefficients. Second, the existence and multiplicity of spatially homogeneous/nonhomogeneous non-constant steady-states are studied by using the Lyapunov–Schmidt reduction. Finally, some numerical simulations are carried out to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

We consider an autonomous ordinary differential equation that admits a heteroclinic loop. The unperturbed heteroclinic loop consists of two degenerate heteroclinic orbits γ 1 and γ 2 . We assume the variational equation along the degenerate heteroclinic orbit γ i has d i ( d i > 1 , i = 1 , 2) linearly independent bounded solutions. Moreover, the splitting indices of the unperturbed heteroclinic orbits are s and − s (s ≥ 0) , respectively. In this paper, we study the persistence of the heteroclinic loop under periodic perturbation. Using the method of Lyapunov-Schmidt reduction and exponential dichotomies, we obtained the bifurcation function, which is defined from R d 1 + d 2 + 2 to R d 1 + d 2 . Under some conditions, the perturbed system can have a heteroclinic loop near the unperturbed heteroclinic loop. [ABSTRACT FROM AUTHOR]

In this paper, we mainly investigate a Lotka-Volterra competition-diffusion-advection system with time delay, where the diffusion and advection rates of two competitors are different. By employing the Lyapunov-Schmidt reduction method, we obtain the existence of steady state solution. A weighted inner product has been introduced to study stability and Hopf bifurcation at the spatially nonhomogeneous steady-state. Our results imply that the infinitesimal generator associated with the linearized system have two pairs of purely imaginary eigenvalues, and time delay can make the spatially nonconstant positive steady state unstable for a reaction-diffusion-advection model. In addition, the bifurcation direction and stability of Hopf bifurcating periodic orbits was obtained by means of the center manifold reduction and the normal form theory. [ABSTRACT FROM AUTHOR]

This article deals with the following fractional Kirchhoff problem with critical exponent a+b∫RN∣(−Δ)s2u∣2dx(−Δ)su=(1+εK(x))u2s∗−1,inRN,\left(a+b\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}| {\left(-\Delta )}^{\tfrac{s}{2}}u\hspace{-0.25em}{| }^{2}{\rm{d}}x\right){\left(-\Delta )}^{s}u=\left(1+\varepsilon K\left(x)){u}^{{2}_{s}^{\ast }-1},\hspace{1.0em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{N}, where a,b>0a,b\gt 0 are given constants, ε\varepsilon is a small parameter, 2s∗=2NN−2s{2}_{s}^{\ast }=\frac{2N}{N-2s} with 04sN\gt 4s, i.e., we show that there exist two nondegenerate positive solutions which seem to be completely different from the result of the fractional Schrödinger equation or the low-dimensional fractional Kirchhoff equation. Using the finite-dimensional reduction method and perturbed arguments, we also obtain the existence of positive solutions to the singular perturbation problems for ε\varepsilon small.

This paper performs an in-depth qualitative analysis of the dynamic behavior of a diffusive SIS epidemic system with delay in heterogeneous environment subject to homogeneous Neumann boundary condition. Firstly, we explore the principal eigenvalue to obtain the stability of the disease-free equilibrium (DFE) and the effect of the nonhomogeneous coefficients on the stable region of the DFE. Secondly, we obtain the existence, multiplicity and explicit structure of the endemic equilibrium (EE), i.e., spatially nonhomogeneous steady-state solutions, by using the implicit function theorem and Lyapunov-Schmidt reduction method. Furthermore, by analyzing the distribution of eigenvalues of infinitesimal generators, the stability of EE and the existence of Hopf bifurcations at EE are given. Finally, the direction of Hopf bifurcation and stability of the bifurcating periodic solution are obtained by virtue of normal form theory and center manifold reduction. [ABSTRACT FROM AUTHOR]

We consider the following problem with combined nonlinearities: { − Δ u + a (y) u + b (y) u q − u p = 0 , u > 0 , in R N , u ∈ H 1 (R N) , where 1 < p < 2 ∗ − 1 , q > 1 , 2 ∗ = 2 N N − 2 when N ≥ 3 ; 2 ∗ = + ∞ when N = 2. By use of the Lyapunov-Schmidt reduction argument, under some expansion conditions of the potentials a (y) and b (y) , we establish infinitely many non-radial solutions. Especially, the nonlinearity b (y) u q is allowed to be (super-)critical, that is, q ≥ 2 ∗ − 1. [ABSTRACT FROM AUTHOR]

We are concerned with the question of constructing a new type of solution to the problem with χ (2) nonlinearities 0.1 { − Δ u + P (x) u = α u v , in R N , − Δ v + Q (x) v = α 2 u 2 + β v 2 , in R N , where P (x) = P (| x |) and Q (x) = Q (| x |) are positive bounded radial potentials, 3 ≤ N < 6 , α > 0 and α > β . Assuming that the potentials P (x) and Q (x) satisfy certain conditions, the existence of a new type of solutions is proved. [ABSTRACT FROM AUTHOR]

We investigate the existence and branching patterns of wave trains (also called periodic traveling waves) in a 2D face-centered square lattice consisting of alternating light and heavy atoms, with linear coupling between nearest particles and a nonlinear substrate potential. In contrast to monatomic chains, we consider two different periodic waveform functions corresponding to light and heavy particles, respectively. As a result, we have to solve coupled advanced-delay differential equations, which are reduced to a finite-dimensional bifurcation equation with an inherited Hamiltonian structure by applying a Lyapunov–Schmidt reduction. Then we obtain the small-amplitude solutions in the Hamiltonian system near equilibria in nonresonance and p : q resonance, respectively. In particular, the results can be applied to some one-dimensional diatomic lattices. [ABSTRACT FROM AUTHOR]

This paper deals with the following fractional elliptic equation with critical exponent \[ \begin{cases} \displaystyle (-\Delta)^{s}u=u_{+}^{2_{s}^{*}-1}+\lambda u-\bar{\nu}\varphi_{1}, & \text{in}\ \Omega,\\ \displaystyle u=0, & \text{in}\ {{\mathfrak R}}^{N}\backslash \Omega, \end{cases}\] where $\lambda$ , $\bar {\nu }\in {{\mathfrak R}}$ , $s\in (0,1)$ , $2^{*}_{s}=({2N}/{N-2s})\,(N>2s)$ , $(-\Delta)^{s}$ is the fractional Laplace operator, $\Omega \subset {{\mathfrak R}}^{N}$ is a bounded domain with smooth boundary and $\varphi _{1}$ is the first positive eigenfunction of the fractional Laplace under the condition $u=0$ in ${{\mathfrak R}}^{N}\setminus \Omega$. Under suitable conditions on $\lambda$ and $\bar {\nu }$ and using a Lyapunov-Schmidt reduction method, we prove the fractional version of the Lazer-McKenna conjecture which says that the equation above has infinitely many solutions as $|\bar \nu | \to \infty$. [ABSTRACT FROM AUTHOR]

This paper is concerned with a prey–predator model with population flux by attractive transition. Our previous paper (Oeda and Kuto, 2018, Nonlinear Anal. RWA, 44, 589–615) obtained a bifurcation branch (connected set) of coexistence steady states which connects two semitrivial solutions. In Oeda and Kuto (2018, Nonlinear Anal. RWA, 44, 589–615), we also showed that any positive steady-state approaches a positive solution of either of two limiting systems, and moreover, one of the limiting systems is an equal diffusive competition model. This paper obtains the bifurcation structure of positive solutions to the other limiting system. Moreover, this paper implies that the global bifurcation branch of coexistence states consists of two parts, one of which is a simple curve running in a tubular domain near the set of positive solutions to the equal diffusive competition model, the other of which is a connected set characterized by positive solutions to the other limiting system. [ABSTRACT FROM AUTHOR]

We prove a result characterizing conditions for the existence and uniqueness of solutions of a certain Diophantine equation, then using techniques from equivariant bifurcation theory, we apply the result to prove symmetric Hopf bifurcation type theorems for both dissipative and non-dissipative autonomous wave equations, for a large set of spatial dimensions. For the latter only the classical implicit function theorem is used. The set of admissible spatial dimensions is the union of the perfect squares together with finitely many non-perfect squares. [ABSTRACT FROM AUTHOR]

This paper deals with the existence of multi-bump solutions for the following Kirchhoff equation - a + b ∫ R 3 | ∇ u | 2 d x Δ u + u = (1 - ε q (x)) | u | p - 2 u , x ∈ R 3 , where a , b > 0 , 2 < p < 6 and q (x) ∈ C (R 3 , R +) satisfies some suitable conditions. By using the Lyapunov–Schmidt reduction method, we extend the results in [14] to the Kirchhoff problem. [ABSTRACT FROM AUTHOR]

This paper is concerned with a size-structured diffusive predator–prey model with chemotaxis mechanism. The existence, linearized stability and monotonicity with respect to the growth rates of boundary steady-state solutions are analyzed. Moreover, the global stability of trivial steady-state solution under certain conditions is proved by constructing Lyapunov functional. We investigate the local and global bifurcations of positive steady-state solutions that emanate from semi-trivial steady-state solutions using Lyapunov–Schmidt reduction and bifurcation techniques when the fertility intensity of a predator or prey is used as a bifurcation parameter. It is shown that the nonlinear nonlocal chemotaxis term can lead to the emergence of Allee effect. [ABSTRACT FROM AUTHOR]

We consider rotating solutions of the generalized surface quasi-geostrophic (gSQG) equation, and prove the existence of five relative stationary points, namely, the Lagrange points near a co-rotating point vortex pair. We also construct a branch of solutions locally bifurcating from the point vortex pair, which consist of a regular vortex pair and several point vortices located near those Lagrange points. This result provides a new phenomenon in fluid dynamics similar to the motion of celestial bodies. [ABSTRACT FROM AUTHOR]

In this paper, we consider the following two-component Schrödiner system \[ \begin{cases} -\Delta u_1 +V_1(x) u_1= \mu_1 u_1^{3} +\beta u_1u_2^{2} & \text{in}\;\mathbb{R}^{3},\\ -\Delta u_2+V_2(x) u_2= \mu_2 u_2^{3} +\beta u_1^{2}u_2 & \text{in}\;\mathbb{R}^{3}. \end{cases} \] First, we revisit the proof of the existence of an unbounded sequence of non-radial positive vector solutions of synchronized type obtained in S. Peng and Z. Wang [Segregated and synchronized vector solutions for nonlinear Schrödinger systems, Arch. Rational Mech. Anal. 208 (2013), 305–339] to give a point-wise estimate of the solutions. Taking advantage of these estimates, we then show a non-degeneracy result of the synchronized solutions in some suitable symmetric space by use of the locally Pohozaev identities. The main difficulties of BEC systems come from the interspecies interaction between the components, which never appear in the study of single equations. The idea used to estimate the coupling terms is inspired by the characterization of the Fermat points in the famous Fermat problem, which is the main novelty of this paper. [ABSTRACT FROM AUTHOR]

Let Σ be a surface of constant mean curvature in R 3 with multiple Delaunay ends. Assuming that Σ is non degenerate in this paper we construct new solutions to the Cahn-Hilliard equation ε Δ u + ε − 1 u (1 − u 2) = ℓ ε in R 3 such that as ε → 0 the zero level set of u ε approaches Σ. Moreover, on compacts of the connected components of R 3 ∖ Σ we have 1 − | u ε | → 0 uniformly. [ABSTRACT FROM AUTHOR]

In this paper, we establish the nondegeneracy of positive solutions to the fractional Kirchhoff problem (a + b ∫ R N | (- Δ) s 2 u | 2 d x) (- Δ) s u + u = u p , in R N , where a , b > 0 , 0 < s < 1 , 1 < p < N + 2 s N - 2 s and (- Δ) s is the fractional Laplacian. In particular, we prove that uniqueness breaks down for dimensions N > 4 s , i.e., we show that there exist two non-degenerate positive solutions which seem to be completely different from the result of the fractional Schrödinger equation or the low dimensional fractional Kirchhoff equation. As one application, combining this nondegeneracy result and Lyapunov-Schmidt reduction method, we can derive the existence of solutions to the singularly perturbation problems. [ABSTRACT FROM AUTHOR]

We reduce solution of the Dirichlet problem (x ∈ D ⊂ Rm) Δu(x) + a(x)u(x) = f (x) in D, u = 0 on ∂D to iterative solution of a simpler problem Δu = f (x) in D, u = 0 on ∂D, for which one can use either Fourier series or Green's function method. The method is suitable for numerical computations, particularly when one uses Newton's method for semilinear problems Δu + g(x, u) = 0 in D, u = 0 on ∂D, in dimensions m ≥ 3. [ABSTRACT FROM AUTHOR]

We study the existence of sign-changing solutions for a non-local version of the sinh-Poisson equation on a bounded one-dimensional interval I, under Dirichlet conditions in the exterior of I. This model is strictly related to the mathematical description of galvanic corrosion phenomena for simple electrochemical systems. By means of the finite-dimensional Lyapunov–Schmidt reduction method, we construct bubbling families of solutions developing an arbitrarily prescribed number sign-alternating peaks. With a careful analysis of the limit profile of the solutions, we also show that the number of nodal regions coincides with the number of blow-up points.

Abstract The present paper deals with a class of nonlocal problems. Under some suitable assumptions on the decay rate of the coefficients, we derive the existence of infinitely many positive solutions to the problem by applying reduction method. Comparing to the previous work, we encounter some new challenges because of competing potentials. By doing some delicate estimates for the competing potentials, we overcome the difficulties and find infinitely many positive solutions.

There have been several existence results for the standing waves of FitzHugh-Nagumo equations. Such waves are the connecting orbits of an autonomous second-order Lagrangian system and the corresponding kinetic energy is an indefinite quadratic form in the velocity terms. When the system has two stable hyperbolic equilibria, there exist two stable standing fronts, which will be used in this paper as building blocks, to construct stable standing waves with multiple fronts in case the equilibria are of saddle-focus type. The idea to prove existence is somewhat close in spirit to [6] ; however several differences are required in the argument: facing a strongly indefinite functional, we need to perform a nonlocal Lyapunov-Schmidt reduction; in order to justify the stability of multiple front standing waves, we rely on a more precise variational characterization of such critical points. Based on this approach, both stable and unstable standing waves are found. [ABSTRACT FROM AUTHOR]