An Elliptic Neumann Problem with Subcritical Nonlinearity

Blowing up solutions for an elliptic Neumann problem with sub - or supercritical nonlinearity Part I : N 1⁄4 3

We consider the subor supercritical Neumann elliptic problem Du þ mu 1⁄4 u5þe; u40 in O; @u @n 1⁄4 0 on @O; O being a smooth bounded domain in R ; m40 and ea0 a small number. Hm denoting the regular part of the Green’s function of the operator Dþ m in O with Neumann boundary conditions, and jmðxÞ 1⁄4 m 1 2 þ Hmðx; xÞ; we show that a nontrivial relative homology between the level sets jm and j b...

A Semilinear Fourth Order Elliptic Problem with Exponential Nonlinearity

We study a semilinear fourth order elliptic problem with exponential nonlinearity. Motivated by a question raised in [Li], we partially extend known results for the corresponding second order problem. Several new difficulties arise and many problems still remain to be solved. We list the ones we feel particularly interesting in the final section. Mathematics Subject Classification: 35J65; 35J40.

Compactness along the first branch of unstable solutions for an elliptic problem with a singular nonlinearity

We prove compactness properties along the branches of semi-stable and unstable solutions –whose Morse index is at most one– of the Dirichlet boundary value problem −∆u = λf(x) (1−u)2 on a bounded domain Ω ⊂ R , provided 1 ≤ N ≤ 7. As a byproduct, we are able to follow the second branch of the bifurcation diagram and prove the existence of a second solution for λ in a natural range. We also show...

Arbitrary number of positive solutions for an elliptic problem with critical nonlinearity

We show that the critical nonlinear elliptic Neumann problem ∆u− μu+ u = 0 in Ω, u > 0 in Ω and ∂u ∂ν = 0 on ∂Ω where Ω is a bounded and smooth domain in R, has arbitrarily many solutions, provided that μ > 0 is small enough. More precisely, for any positive integer K, there exists μK > 0 such that for 0 < μ < μK , the above problem has a nontrivial solution which blows up at K interior points ...

Entropy Solutions for Nonlinear Elliptic Anisotropic Homogeneous Neumann Problem