Singularly perturbed Dirichlet problems with subquadratic nonlinearities

Dirichlet Problems with Asymmetric Nonlinearities

Convergence Estimates for Abstract Second Order Singularly Perturbed Cauchy Problems with Monotone Nonlinearities∗

We study the behavior of solutions to the problem  ε ( u′′ ε (t) +A1uε(t) ) + uε(t) +A0uε(t) +B ( uε(t) ) = fε(t), t ∈ (0, T ), uε(0) = u0ε, u ′ ε(0) = u1ε, in the Hilbert space H as ε → 0, where A1, A0 are two linear selfadjoint operators and B is a locally Lipschitz and monotone operator. ∗Accepted for publication in revised form on June 15, 2012. †Department of Mathematics and Informatics,...

Singularly perturbed Neumann problems with potentials

Such a problem was intensively studied in several works. For example, Ni & Takagi, in [11, 12], show that, for ε sufficiently small, there exists a solution uε of (2) which concentrates in a point Qε ∈ ∂Ω andH(Qε) → max∂ΩH , here H denotes the mean curvature of ∂Ω. Moreover in [10], using the LiapunovSchmidt reduction, Li constructs solutions with single peak and multi-peaks on ∂Ω located near ...

Fem for Singularly Perturbed Problems

We consider the numerical approximation of singularly perturbed elliptic boundary value problems over non-smooth domains. We use a decomposition of the solution that contains a smooth part, a corner layer part and a boundary layer part. Explicit guidelines for choosing mesh-degree combinations are given that yield nite element spaces with robust approximation properties. In particular, we const...

Perturbed Subcritical Dirichlet Problems with Variable Exponents

We study a class of nonhomogeneous elliptic problems with Dirichlet boundary condition and involving the p(x)-Laplace operator and powertype nonlinear terms with variable exponent. The main results of this articles establish sufficient conditions for the existence of nontrivial weak solutions, in relationship with the values of certain real parameters. The proofs combine the Ekeland variational...