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In this paper, we show finite time blow-up of solutions of the p−wave
equation in ℝN, with critical Sobolev exponent. Our work
extends a result by Galaktionov and Pohozaev [4]
Let be a bounded domain in with smooth boundary . We consider the equation , under zero Neumann boundary conditions, where is open, smooth and bounded and is a small positive parameter. We assume that there is a -dimensional closed, embedded minimal submanifold of , which is non-degenerate, and certain weighted average of sectional curvatures of is positive along . Then we prove the existence of a sequence and a positive solution such that in the sense of measures, where ...
Solutions to nonlinear Schrödinger equations may blow up in finite time. We study the influence of the introduction of a potential on this phenomenon. For a linear potential (Stark effect), the blow-up time remains unchanged, but the location of the collapse is altered. The main part of our study concerns isotropic quadratic potentials. We show that the usual (confining) harmonic potential may anticipate the blow-up time, and always does when the power of the nonlinearity is -critical. On the other...
Motivated by a problem arising in astrophysics we study a nonlinear elliptic equation in RN with cylindrical symmetry and with singularities on a whole subspace of RN. We study the problem in a variational framework and, as the nonlinearity also displays a critical behavior, we use some suitable version of the Concentration-Compactness Principle. We obtain several results on existence and nonexistence of solutions.
We classify the global behavior of weak solutions of the Keller-Segel system of degenerate and nondegenerate type. For the stronger degeneracy, the weak solution exists globally in time and has a uniform time decay under some extra conditions. If the degeneracy is weaker, the solution exhibits a finite time blow up if the data is nonnegative. The situation is very similar to the semilinear case. Some additional discussion is also presented.
We consider the Yamabe type family of problems , in , on , where is an annulus-shaped domain of , , which becomes thinner as . We show that for every solution , the energy
as well as the Morse
index tend to infinity as . This is proved through a fine blow up analysis of appropriate scalings of solutions whose limiting profiles are regular, as well as of singular solutions of some elliptic problem on , a half-space or an infinite strip. Our argument also involves a Liouville type
theorem...
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