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

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

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...

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

Bifurcation of Positive Solutions for a Semilinear Equation with Critical Sobolev Exponent

In this note we consider bifurcation of positive solutions to the semilinear elliptic boundary-value problem with critical Sobolev exponent −∆u = λu− αu + u −1, u > 0, in Ω, u = 0, on ∂Ω. where Ω ⊂ Rn, n ≥ 3 is a bounded C2-domain λ > λ1, 1 < p < 2∗ − 1 = n+2 n−2 and α > 0 is a bifurcation parameter. Brezis and Nirenberg [2] showed that a lower order (non-negative) perturbation can contribute t...

Radially Symmetric Solutions for a Class ofCritical Exponent Elliptic Problems in RN

We give a method for obtaining radially symmetric solutions for the critical exponent problem ?u + a(x)u = u q + u 2 ?1 in R N u > 0 and R R N jruj 2 < 1 where, outside a ball centered at the origin, the non-negative function a is bounded from below by a positive constant ao > 0. We remark that, diierently from the literature, we do not require any conditions on a at innnity.