Multiple positive solutions for singularly perturbed elliptic problems in exterior domains

Multiple Positive Solutions of Semilinear Elliptic Problems in Exterior Domains

Multiple Solutions for Singularly Perturbed Semilinear Elliptic Equations in Bounded Domains

We are concerned with the multiplicity of solutions of the following singularly perturbed semilinear elliptic equations in bounded domains Ω:−ε2∆u+ a(·)u= u|u|p−2 in Ω, u > 0 in Ω, u= 0 on ∂Ω. The main purpose of this paper is to discuss the relationship between the multiplicity of solutions and the profile of a(·) from the variational point of view. It is shown that if a has a “peak” in Ω, the...

Multiple Boundary Peak Solutions for Some Singularly Perturbed Neumann Problems

We consider the problem " 2 u ? u + f (u) = 0 in u > 0 in ; @u @ = 0 on @; where is a bounded smooth domain in R N , " > 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 " approaches zero, at a critical point of the mean curvature function H(P); P 2 @. It is also known ...

Heterogeneous Domain Decomposition for Singularly Perturbed Elliptic Boundary Value Problems

A heterogeneous domain-decomposition method is presented for the numerical solution of singularly perturbed elliptic boundary value problems. The method, which is parallelizable at various levels, uses several ideas of asymptotic analysis. The subdomains match the domains of validity of the local [ “inner” and “outer”) asymptotic expansions, and cut-off functions are used to match solutions in ...

Symmetry of Nodal Solutions for Singularly Perturbed Elliptic Problems on a Ball

In [40], it was shown that the following singularly perturbed Dirichlet problem 2∆u− u+ |u|p−1u = 0, in Ω, u = 0 on ∂Ω has a nodal solution u which has the least energy among all nodal solutions. Moreover, it is shown that u has exactly one local maximum point P 1 with a positive value and one local minimum point P 2 with a negative value and, as → 0, φ(P 1 , P 2 ) → max (P1,P2)∈Ω×Ω φ(P1, P2), ...