On a Semilinear Elliptic Equation in R when the Exponent Approaches Infinity

On a Semilinear Elliptic Equation in R 2 When the Exponent Approaches Innnity

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...

A two-phase free boundary problem for a semilinear elliptic equation

In this paper we study a two-phase free boundary problem for a semilinear elliptic equation on a bounded domain $Dsubset mathbb{R}^{n}$ with smooth boundary‎. ‎We give some results on the growth of solutions and characterize the free boundary points in terms of homogeneous harmonic polynomials using a fundamental result of Caffarelli and Friedman regarding the representation of functions whose ...

The Nehari Manifold for a Class of Indefinite Weight Semilinear Elliptic Equations

EXISTENCE OF A STEADY FLOW WITH A BOUNDED VORTEX IN AN UNBOUNDED DOMAIN

We prove the existence of steady 2-dimensional flows, containing a bounded vortex, and approaching a uniform flow at infinity. The data prescribed is the rearrangement class of the vorticity field. The corresponding stream function satisfies a semilinear elliptic partial differential equation. The result is proved by maximizing the kinetic energy over all flows whose vorticity fields are rearra...

Nonexistence of Nonconstant Global Minimizers with Limit at ∞ of Semilinear Elliptic Equations in All of Rn

We prove nonexistence of nonconstant global minimizers with limit at infinity of the semilinear elliptic equation −∆u = f(u) in the whole R , where f ∈ C(R) is a general nonlinearity and N ≥ 1 is any dimension. As a corollary of this result, we establish nonexistence of nonconstant bounded radial global minimizers of the previous equation.