Juncheng Wei

Multiple boundary peak solutions for some singularly perturbed Neumann problems

Abstract We consider the problem e 2 Δu−u+f(u)=0 in Ω, u>0 in Ω, ∂u/∂ν=0 on ∂Ω, where Ω is a bounded smooth domain in R N , e>0 is a small parameter and f is a superlinear, subcritical nonlinearity. It is known that this equation possesses boundary spike solutions such that the spike concentrates, as e approaches zero, at a critical point of the mean curvature function H(P),P∈∂Ω . It is also known that this equation has multiple boundary spike solutions at multiple nondegenerate critical points of H(P) or multiple local maximum points of H(P). In this paper, we prove that for any fixed positive integer K there exist boundary K-peak solutions at a local minimum point of H(P). This implies that for any smooth and bounded domain there always exist boundary K-peak solutions. We first use the Liapunov–Schmidt method to reduce the problem to finite dimensions. Then we use a maximizing procedure to obtain multiple boundary spikes.

Multiple interior peak solutions for some singularly perturbed neumann problems

We consider the problem { e 2 Δ u − u + f ( u ) = 0 , u > 0 , in Ω , ∂ u ∂ v = 0 on ∂ , where Ω is a bounded smooth domain in RN, ɛ > 0 is a small parameter, and ƒ is a superlinear, subcritical nonlinearity. It is known that this equation possesses boundary spike solutions that concentrate, as ɛ approaches zero, at a critical point of the mean curvature function H(P), P e ∂Ω. It is also proved that this equation has single interior spike solutions at a local maximum point of the distance function d(P, ∂Ω), P e Ω. In this paper, we prove the existence of interior K −peak (K ⩾ 2) solutions at the local maximum points of the following function φ(P1, P2,…, pk) = mini, k, l = 1, …, K; k ≠ 1 (d(Pi, ∂Ω), 1/2 ¦Pk −pl¦). We first use the Liapunov-Schmidt reduction method to reduce the problem to a finite dimensional problem. Then we use a maximizing procedure to obtain multiple interior spikes. The function φ(P1, …, pk) appears naturally in the asymptotic expansion of the energy functional.

The stability of spike solutions to the one-dimensional Gierer—Meinhardt model

Abstract The stability properties of an N-spike equilibrium solution to a simplified form of the Gierer–Meinhardt activator–inhibitor model in a one-dimensional domain is studied asymptotically in the limit of small activator diffusivity e. The equilibrium solution consists of a sequence of spikes of equal height. The two classes of eigenvalues that must be considered are the O(1) eigenvalues and the O(e2) eigenvalues, which are referred to as the large and small eigenvalues, respectively. The spike pattern is stable when the parameters in the Gierer–Meinhardt model are such that both sets of eigenvalues lie in the left half-plane. For a certain range of these parameters and for N≥2 and e→0, it is shown the O(1) eigenvalues are in the left half-plane only when D D N ∗ , where D N ∗ is another critical value of D, which satisfies D N ∗ N . Thus, when N≥2 and e≪1, the spike pattern is stable only when D N ∗ . An explicit formula for D N ∗ is given. For the special case N=1, it is shown that a one-spike equilibrium solution is stable when D D1(e). An asymptotic formula for D1(e) is given. Finally, the dynamics of a one-spike solution is studied by deriving a differential equation for the trajectory of the center of the spike.

Stationary solutions for the Cahn-Hilliard equation

We study the Cahn-Hilliard equation in a bounded domain without any symmetry assumptions. We assume that the mean curvature of the boundary has a nondegenerate critical point. Then we show that there exists a spike-like stationary solution whose global maximum lies on the boundary. Our method is based on Lyapunov-Schmidt reduction and the Brouwer fixed-point theorem.

On single interior spike solutions of Gierer-Meinhardt system : uniqueness, spectrum estimates and stability analysis

We study the interior spike solutions to a steady state problem of the shadow system of the Gierer–Meinhardt system arising from biological pattern formation. We first show that at a non-degenerate peak point the interior spike solution is locally unique, and then we establish the spectrum estimates of the associated linearized operator. We also prove that the corresponding solution to the shadow system is unstable. Furthermore, the metastability of such solutions is analysed.

ON THE ROLE OF MEAN CURVATURE IN SOME SINGULARLY PERTURBED NEUMANN PROBLEMS

We construct solutions exhibiting a single spike-layer shape around some point of the boundary as

ON A TWO-DIMENSIONAL ELLIPTIC PROBLEM WITH LARGE EXPONENT IN NONLINEARITY

\var \to 0

Convergence for a Liouville equation

for the problem \left\{ \begin{array}{l} \var^2 \tri u - u + u^p = 0 \quad \mbox{in} \ \Omega, \\ u > 0 \quad \mbox{in} \ \Omega, \\ \frac{\partial u}{\partial \nu} = 0 \quad \mbox{on} \ \partial \Omega, \end{array} \right.\label {1.1} where

On the multiplicity of solutions of two nonlocal variational problems

\Omega

On bound states concentrating on spheres for the Maxwell-Schrödinger equation

is a bounded domain with smooth boundary in