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

Instability of Elliptic Equations on Compact Riemannian Manifolds with Non-negative Ricci Curvature

We prove the nonexistence of nonconstant local minimizers for a class of functionals, which typically appear in scalar two-phase field models, over smooth N -dimensional Riemannian manifolds without boundary and nonnegative Ricci curvature. Conversely, for a class of surfaces possessing a simple closed geodesic along which the Gauss curvature is negative, we prove the existence of nonconstant l...

Instability results for an elliptic equation on compact Riemannian manifolds with non-negative Ricci curvature

We prove nonexistence of nonconstant local minimizers for a class of functionals, which typically appears in the scalar two-phase field model, over a smoothN−dimensional Riemannian manifold without boundary with non-negative Ricci curvature. Conversely for a class of surfaces possessing a simple closed geodesic along which the Gauss curvature is negative we prove existence of nonconstant local ...

Regularity of minimizers for three elliptic problems: minimal cones, harmonic maps, and semilinear equations

We discuss regularity issues for minimizers of three nonlinear elliptic problems. They concern minimal cones, minimizing harmonic maps into a hemisphere, and radial local minimizers of semilinear elliptic equations. We describe the strong analogies among the three regularity theories. They all use a method originated in a paper of J. Simons on the area minimizing properties of cones.

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

Existence of Large Solutions to Semilinear Elliptic Equations with Multiple Terms

We consider the semilinear elliptic equation 4u = p(x)uα + q(x)uβ on a domain Ω ⊆ Rn, n ≥ 3, where p and q are nonnegative continuous functions with the property that each of their zeroes is contained in a bounded domain Ωp or Ωq, respectively in Ω such that p is positive on the boundary of Ωp and q is positive on the boundary of Ωq. For Ω bounded, we show that there exists a nonnegative soluti...