Pedro M. Girão

Multibump nodal solutions for an indefinite superlinear elliptic problem

Abstract We define some Nehari-type constraints using an orthogonal decomposition of the Sobolev space H 0 1 and prove the existence of multibump nodal solutions for an indefinite superlinear elliptic problem.

On the Global Uniqueness for the Einstein–Maxwell-Scalar Field System with a Cosmological Constant: Part 3. Mass Inflation and Extendibility of the Solutions

This paper is the third part of a trilogy dedicated to the following problem: given spherically symmetric characteristic initial data for the Einstein–Maxwell-scalar field system with a cosmological constant

Existence and nonexistence of least energy solutions of the Neumann problem for a semilinear elliptic equation with critical Sobolev exponent and a critical lower-order perturbation

\Lambda

Multi-bump nodal solutions for an indefinite non-homogeneous elliptic problem

Λ, with the data on the outgoing initial null hypersurface given by a subextremal Reissner–Nordström black hole event horizon, study the future extendibility of the corresponding maximal globally hyperbolic development as a “suitably regular” Lorentzian manifold. In the first part [7] of this series we established the well posedness of the characteristic problem, whereas in the second part [8] we studied the stability of the radius function at the Cauchy horizon. In this third and final paper we show that, depending on the decay rate of the initial data, mass inflation may or may not occur. When the mass is controlled, it is possible to obtain continuous extensions of the metric across the Cauchy horizon with square integrable Christoffel symbols. Under slightly stronger conditions, we can bound the gradient of the scalar field. This allows the construction of (non-isometric) extensions of the maximal development which are classical solutions of the Einstein equations. Our results provide evidence against the validity of the strong cosmic censorship conjecture when

Bifurcation curves of a diffusive logistic equation with harvesting orthogonal to the first eigenfunction

\Lambda >0

Bifurcation curves of a logistic equation when the linear growth rate crosses a second eigenvalue

Λ>0.