In this work, we study the local wellposedness of solution to a nonlinear elliptic-dispersive coupled system which serves as model for Micro-Electro-Mechanical System (MEMS). A simple electrostatically actuated MEMS capacitor device consists two parallel plates separated by gas-filled thin gap. The modelling combines linear elliptic equation gas pressure with semilinear dispersive gap width. We...

We consider a semilinear elliptic equation in R 2 with the nonlinear exponent approaching innnity. In contrast to the blow-up behavior of the corresponding problem in R n with n 3, the L 1 (R 2) norms of the solutions to the equation in R 2 remain bounded from below and above. After a careful study on the decay rates of several quantities, we prove that the normalized solutions will approach th...

We consider a semilinear elliptic equation on a smooth bounded domain Ω in R2, assuming that both the domain and the equation are invariant under reflections about one of the coordinate axes, say the y-axis. It is known that nonnegative solutions of the Dirichlet problem for such equations are symmetric about the axis, and, if strictly positive, they are also decreasing in x for x > 0. Our goal...

Abstract. We discuss a general framework for the numerical solution of a family of semilinear elliptic problems whose leading differential operator is the Laplacian. A problem is first transformed to one on a standard domain via a conformal mapping. The boundary value problem on the standard domain is then reduced to an equivalent integral operator equation. We employ the Galerkin method to sol...

The semilinear reaction-diffusion equation−ε4u+b(x, u) = 0 with Dirichlet boundary conditions is considered in a convex unbounded sector. The diffusion parameter ε is arbitrarily small, and the “reduced equation” b(x, u0(x)) = 0 may have multiple solutions. A formal asymptotic expansion for a possible solution u is constructed that involves boundary and corner layer functions. For this asymptot...

We consider a semilinear elliptic equation in R with the nonlinear exponent approaching infinity. In contrast to the blow-up behavior of the corresponding problem in Rn with n ≥ 3, the L(R) norms of the solutions to the equation in R remain bounded from below and above. After a careful study on the decay rates of several quantities, we prove that the normalized solutions will approach the funda...