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Perspectives in Nonlinear Partial Differential Equations: In by Henri Berestycki, Michiel Bertsch, Felix E. Browder, Louis

By Henri Berestycki, Michiel Bertsch, Felix E. Browder, Louis Nirenberg, Lambertus A. Peletier

In get together of Haim Brezis's sixtieth birthday, a convention used to be held on the Ecole Polytechnique in Paris, with a software attesting to Brezis's wide-ranging impact on nonlinear research and partial differential equations. The articles during this quantity are basically from that convention. They current an extraordinary view of the state-of-the-art of many elements of nonlinear PDEs, in addition to describe new instructions which are being spread out during this box. The articles, written via mathematicians on the middle of present advancements, supply a little extra own perspectives of the real advancements and demanding situations.

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Extra resources for Perspectives in Nonlinear Partial Differential Equations: In Honor of Haïm Brezis

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2. The case of potentials bounded away from zero In this section we shortly outline the case in which V is smooth and satisfies :3 a , b > 0 ( ��n, such that a :::; V(x) :::; b. We also take p E 1, and K (x) > 0 smooth and bounded. In order to prove the existence of solution to (N LB,:) it is convenient to introduce the auxiliary potential 1 �. () = p + Q (x) = VO (x)K -2/ (P - I) (x), p-1 2 The role of Q is highlighted by the fact that a necessary condition for the concen­ tration of solutions of (N LB,,J at a point Xo is that Q' (xo) = o.

This conjecture has been verified for n = 2 (when I: is a curve) by Del Pino, Kowalczyk and Wei in [15J . As mentioned in Remark 6-(iii), from the solutions concentrating on a sphere branch off solutions with a different profile. We suspect that these solutions oscillate along the sphere and that, following the bifurcation branches, they concentrate on points or on lower dimensional manifolds. This bifurcation analysis should be pursued also for solutions considered in the previous item. In [16J the authors studied, for n = and < < 5, the case in which V is negative in some interval (or intervals) of R They are able to produce, when € � 0, solutions of (NLS,;;) (for K == 1) which are highly oscillatory in {V :::; O}, and which decay exponentially away from this region.

We do not have much information though, on the zeroes located outside the support of a, the " invisible vortices" . 32). One may hope to prove such a �esult by an analogue of a Newton method. Our results deal with an upper bound for the energy. A natural question would be to get also the lower bound and prove r convergence type results. 17) does not contain the coefficient b = ,(j ) . 8. T h1 /4. 33) provides a lower bound for the energy. 9) requires a lot of vortices and thus creates a contribution in the energy through b.

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