Workshop on Variational and Topological Methods in the Study of Nonlinear Problems

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2 Workshop on Variational and Topological Methods in the Study of Nonlinear Problems June 29 July 1, 2015 Practical Information Workshop Website: The hotel for the participants is Zenitude Besançon - La City, 11 avenue Louise Michel, Besançon. Website: besancon-la-city.htm The workshop will take place at the Centre Diocésain located in the town center of Besançon. Its address is 20 rue Mégevand, Besançon. Note that the town center is very small. Hence, walking is probably the best option to reach the conference from your hotel. The tramway/bus stops which are the closest to the Centre Diocésain are : tramway stop Chamars, bus stop Mégevand, bus stop Granvelle and bus stop Mairie. Further informations can be found at se-deplacer/plans/cartographie-interactive To arrive at the Centre Diocésain from the Besançon Viotte station, you can take tramway number 2 and reach the Chamars tramway stop (10 mn). Then follow the following map (5-10 mn walk). You can also take the bus number 5 and reach the bus stop Mairie. 2

3 Some restaurants that we know Open Open on on On the map Name Address Phone Food/Cuisine Prices Remarks Sunday Monday night night 41 (C4) Brasserie Rue Lacoré, Place Granvelle Yes Yes Regional french cuisine Menu : 28,50 euros Large dining room 42 (B2) Chez Achour 77, Rue Battant Yes No North african cuisine Menus : 21 and 28,90 euros 6 (A2) Da Gianni 9 Rue Richebourg No Yes Italian Pizzas & Pasta from 9 to 22 Le Kaf Des Vieux 21 (C3) 48 Rue François Louis Bersot Yes Yes Regional french cuisine Menus : 19 and 27 euros Amants L'Affineur 5 (B2) 82 Rue Battant No Yes Cheese specialities Meals ~15 euros Comtois 43 (B3) L'Annexe 11 Rue Du Palais De Justice No No French cuisine Menu : 34 euros 15 (B2) La Grange 17 Avenue Elisée Cusenier Yes Yes Cheese specialities Meals around euros Bio traditional french 44 (B3) La Mirabelle 1 Rue Megevand No Yes Meals from 15 to 30 euros cuisine Betw. 9 and 21 (C3)La Papaye Verte 28 Rue François Louis Bersot No No Vietnamese Menus from 20 to 30. 'real-time' cooking, slower service 9 (C3) Le Barthod Rue Bersot No No French cuisine 19 (B2) Le Champagney 37 Rue Battant No No French cuisine Menu : 38 euros Menus : 29 and 41 euros 45 (B3) Le Coucou 12 Rue Luc Breton No No Cheese specialities Meals around 15 euros Small restaurant 10 (C3) Le Lotus D'Or 78 Rue Des Granges No Yes Cambodian Menus from 16 to 18 euros 46 (C3) Le Poker D'as 14 Square Saint-Amour No No French cuisine Menus from 23,50 to 55 euros

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6 Workshop on Varia-onal and Topological Methods in the Study of Nonlinear Problems Besançon, Centre Diocésain, June 29 July 1, 2015 Monday, June 29 Tuesday, June 30 Wednesday, July 1 9:30-10:15 Coffee 9:00-9:50 Tobias Weth 9:00-9:50 Jacques Giacomoni 10:15-10:30 WELCOME 10:00-10:30 Coffee 10:00-10:30 Coffee 10:30-11:20 Boris Buffoni 10:30-11:20 ColeQe De Coster 10:30-11:20 Kazunaga Tanaka 11:30-12:20 Maria Esteban 11:30-12:20 Huan- Song Zhou 11:30-12:20 Silvia Cingolani 12:30-14:00 Lunch 12:30-14:00 Lunch 12:30-14:00 Lunch 14:00-14:50 David Arcoya 14:00-14:50 Charles Stuart 14:00-14:50 Marco Squassina 15:00-15:50 Denis Bonheure 15:00-15:50 Henri Berestycki 15:00-15:50 Jean Mawhin 16:00-16:30 Coffee 16:00-16:30 Coffee 16:30-17:20 Piero Montecchiari 16:30-17:20 Andrzej Szulkin Conferences, lunches and coffees are in Centre Diocésain 19:45 Social dinner ( Restaurant Saint Pierre )

7 David Arcoya (Universidad de Granada) Elliptic problems with regularizing terms We show [1] that the presence of some terms has a regularizing effect on the solution of some nonlinear Dirichlet problems. The simplest example is the linear problem { div (M(x) u) + a(x) u = f(x), x Ω, u = 0, x Ω, where Ω is a bounded open set of R N, M is a bounded elliptic matrix and 0 a(x) L 1 (Ω). Even if f(x) only belongs to L 1 (Ω), the assumption there exists Q > 0 such that f(x) Q a(x) implies the existence of a weak solution u belonging to W 1,2 0 (Ω) and to L (Ω). [1] D. Arcoya, L. Boccardo, Regularizing effect of the interplay between coefficients in some elliptic equations, Journal of Functional Analysis, 268, (2015), Henri Berestycki (EHESS, Paris) The effect of a line with fast diffusion on Fisher-KPP reaction-diffusion equations I will present a system of equations describing the effect of inclusion of a line (the road ) with fast diffusion on biological invasions in the plane (the field ). Outside of the road, the propagation is of the classical Fisher-KPP type. We find that past a certain threshold for the ratio of diffusivity coefficients, the presence of the road enhances the speed of global propagation. I will discuss several further effects such as transport or reaction on the road. I will also derive the asymptotic behaviour of the invasion speed and the expansion shape depending on the various parameters. I report here on results from a series of joint works with Jean-Michel Roquejoffre and Luca Rossi. Denis Bonheure (Université Libre de Bruxelles) Existence and symmetry of least energy nodal solutions for Hamiltonian elliptic systems In this talk, I will discuss the existence of least energy nodal solutions for the Hamiltonian elliptic system with Hénon type weights u = x β v q 1 v, v = x α u p 1 u in Ω, u = v = 0 on Ω, where Ω is a bounded smooth domain in R N, N 1, α, β 0 and the nonlinearities are superlinear and subcritical, namely 1 > 1 p q + 1 > N 2 N. When Ω is either a ball or an annulus centred at the origin and N 2, we show that these solutions display the so-called foliated Schwarz symmetry. It is natural to conjecture that these solutions are not radially symmetric. We provide such a symmetry breaking in a range of parameters where the solutions of the system behave like the solutions of a single equation. Our results on the above system are new even in the case of the Lane-Emden system (i.e. without weights). As far as we know, these are the first results about least energy nodal solutions for strongly coupled elliptic systems and their symmetry properties. 7

8 Boris Buffoni (EPFL, Lausanne) On the existence of fully localized surface waves with weak surface tension Lumps are fully localized waves travelling at the upper surface of a three-dimensional layer of perfect irrotational fluid. They were observed numerically in presence of gravity and surface tension when the density is constant and the depth finite. Their existence was also proved mathematically when in addition the surface tension is strong. The aim is to extend the existence result to the regime of weak surface tension. This is work in progress jointly with M. D. Groves and E. Wahlén. Silvia Cingolani (Politecnico di Bari) Concentration on circles for magnetic Nonlinear Schrödinger Equations In my talk I will consider the stationary nonlinear Schrödinger equation (i + A(x)) 2 u + V (x)u = u p 2 u, x R 3, (1) where p > 2, A is a vector (magnetic) potential associated to a given magnetic field B, i.e B = A, and V is a nonnegative, scalar (electric) potential which can be singular at the origin and vanish at infinity or outside a compact set. I will present recent results, obtained in a joint work with Denis Bonheure and Manon Nys, proving the existence of semiclassical cylindrically symmetric solutions of (1) whose moduli concentrate, as 0, around circles driven by the magnetic and the electric potentials. Our results show that the magnetic field influences the location (not only the phase) of the semiclassical states when the concentration occurs around a higher dimensional set, not a single point. Colette De Coster (Université de Valenciennes) Existence and multiplicity result for an elliptic problem with critical growth in the gradient In this talk, we consider the boundary value problem (P λ ) u = λc(x)u + µ(x) u 2 + h(x), u H 1 0 (Ω) L (Ω), where Ω R N, N 3 is a bounded domain with smooth boundary. It is assumed that c, h belong to L p (Ω), µ L (Ω) and c 0. In case λ < 0, we prove that the problem (P λ ) has at most one solution and we give condition under which the existence of a solution of (P λ ) is obtained. Moreover these solutions belong to a continuum, the behaviour of which depends in an essential way on the existence of a solution of (P 0 ). The geometry of the set of solutions for λ > 0 is much more complicated. Under the assumption µ µ 1 > 0 for some µ 1 R, we derive informations on this set. In particular we prove multiplicity results as well as existence of positive, negative, non-positive, non-negative solutions. We show also that the situation is completely different for positive or negative h. [1] D. Arcoya, C. De Coster, L. Jeanjean, K. Tanaka, Continuum of solutions for an elliptic problem with critical growth in the gradient, J. Funct. Anal. (2015), [2] D. Arcoya, C. De Coster, L. Jeanjean, K. Tanaka, Remarks on the uniqueness for quasilinear elliptic equations with quadratic growth conditions, J. Math. Anal. Appl., 420, (2014), [3] C. De Coster, L. Jeanjean, Multiplicity results in the non-resonant case for an elliptic problem with critical growth in the gradient, preprint. 8

9 Maria Esteban (Université Paris Dauphine) Optimal symmetry results for the optimizers of Caffarelli- Kohn-Nirenberg inequalities In this talk I will present recent results, obtained in collaboration with J. Dolbeault and M. Loss, proving the radial symmetry of the optimizers of Caffarelli- Kohn-Nirenberg inequalities whenever they are local minima in the full functional space. These results are optimal and close a series of works proving partial results. The method used to obtain this result, which actually proves the uniqueness of positive solutions for the corresponding Euler-Lagrange equations, is based on a nonlinear fast diffusion flow which is applied around any positive solution to explore the energy landscape around it. Jacques Giacomoni (Université de Pau) Uniqueness of positive solutions of n-laplace equation in a ball in R n with exponential nonlinearity Let n 2 and Ω R n be a bounded domain. Then by Trudinger embedding, W 1,n (Ω) is embedded in an Orlicz space consisting of exponential functions. Consider the corresponding semilinear n-laplace equation with critical or sub-critical exponential nonlinearity in a ball B(R) with Dirichlet boundary condition. We prove that under suitable growth conditions on the nonlinearity, the problem admits a unique large and non degenerate positive radial solution u. Jean Mawhin (Université Catholique de Louvain) Periodic solutions of relativistic-type systems with periodic nonlinearities The lecture surveys recent approaches and results on the multiplicity of periodic solutions of differential systems of the type ( u 1 u 2 ) + u V (t, u) = e(t) when the potential V is periodic in each component of u and e has mean value zero over the time period. Piero Montecchiari (Università Politecnica delle Marche) Differently shaped transition solutions for some class of semilinear elliptic equations We consider some class of semilinear elliptic equations u + F u (x, u) = 0, x R n, where F is a periodic function in the space variables x and has the behaviour of a multiple well potential in the variable u. We discuss some global variational approach to study the problem of existence of differently shaped transition solutions. Marco Squassina (Università degli Studi di Verona) Eigenvalues of the fractional p-laplacian and the Brezis-Nirenberg problem We discuss the content of joint works with Lorenzo Brasco (Marseille), Antonio Iannizzotto (Cagliari), Sunra J. Mosconi (Verona), Enea Parini (Marseille), Kanishka Perera (Melbourne) and Yang Yang (Wuxi) about the nonlinear eigenvalues of the fractional p-laplacian and the complete solution to the Brezis-Nirenberg problem in this framework. 9

10 Charles A. Stuart (EPFL, Lausanne) Bifurcation for a critically degenerate elliptic Dirichlet eigenvalue problem Let Ω be an open, bounded subset of R N for N 3 such that 0 Ω. We consider the boundary value problem A(x) u + V (x)u + n(x, u) + g(x, u) = λ{u + h(x, u)} in Ω (1) u = 0 on Ω, (2) where the nonlinear terms are of higher order near 0 in the sense that n(x, ξ) g(x, s) h(x, s) lim = lim = lim = 0 (3) ξ 0 ξ s 0 s s 0 s for all x Ω. The potential V is bounded and the coefficient A satisfies A(x) (A) A C(Ω) with A(x) > 0 for all x 0 and lim x 0 x = a for some t [0, 2] t and a (0, ). For t > 0, the ellipticity degenerates at x = 0 and we concentrate on the case t = 2. We consider solutions of (1),(2) having finite energy, Ω A u 2 dx <, bifurcating from the line of trivial solutions {(λ, 0) : λ R}. The linearization of (1),(2) at the trivial solution (λ, 0) is A(x) u + V (x)u = λu in Ω (4) u = 0 on Ω. (5) For 0 < t < 2, the relation between bifurcation points for (1),(2) and the spectrum of (4),(5) is what one expects from classical bifurcation theory, as in the uniformly elliptic case t = 0. However, the degeneracy when t = 2 is critical and the results are quite different in this case. Following earlier work with Gilles Evéquoz, I shall present some new results. Andrzej Szulkin (Stockholm University) Ground states for problems of Brezis-Nirenberg type with critical and supercritical exponent We consider the elliptic boundary value problem u = λu + u 2 N,k 2 u in Ω, u = 0 on Ω, where 2 N,k := 2(N k)/(n k 2) is the (k + 1)-st critical exponent, Ω Rk+1 R N k 1 is a bounded domain, invariant with respect to the action of O(k + 1) on R k+1 and bounded away from R N k 1. We show that this problem has no nontrivial O(k + 1)-invariant solution for any λ (, λ ), where λ > 0, and that such solutions exist in a left neighbourhood of each O(k + 1)-invariant eigenvalue λ (k) m. Moreover, they are ground states (for solutions with this symmetry) and bifurcate at λ (k) m. This problem is related to the question of existence of bifurcating ground states for the anisotropic equation div(a(x) u) = λb(x)u + c(x) u 2 2 u in Θ, u = 0 on Θ, where Θ R n is bounded, a, b, c > 0 and 2 = 2n/(n 2). This is joint work with Mónica Clapp and Angela Pistoia. 10

11 Kazunaga Tanaka (Waseda University, Tokyo) Singular perturbation problems for NLS systems and nonlinear elliptic problems in perturbed cylindrical domains In this talk we consider singular perturbation problems for a system of NLS: ε 2 u + V 1 (x)u = µ 1 u 3 + βuv 2 in R N, ε 2 v + V 2 (x)v = µ 2 v 3 + βu 2 v in R N and for a nonlinear elliptic problem in perturbed cylindrical domains: u = au + u p in Ω ε, u = 0 on Ω ε, where Ω ε = {(x, y) R k R l ; x R k, y D εx } and D x R l is a domain depending on x R k smoothly. We show the existence of solutions which concentrate at a prescribed part of the domain especially we consider the situation where the prescribed part is corresponding to local maxima. Tobias Weth (Goethe-Universität, Frankfurt) On isoperimetric profiles for the first nontrivial Neumann and Steklov eigenvalues in Riemannian manifolds We consider the geometric variational problems of maximizing the first nontrivial Neumann and Steklov eigenvalues of the Laplace-Beltrami operator among subdomains of a Riemannian manifold under a fixed volume constraint. We will mainly be concerned with the corresponding local isoperimetric (or, more precisely, isochoric) profiles. As a corollary of our analysis, we deduce local isoperimetric comparison principles depending only on the scalar curvature. Huan-Song Zhou (Wuhan Institute of Physics and Mathematics) Concentration behavior and symmetry breaking for Gross-Pitaevskii equation with ring-shaped potential This talk is concerned with the properties of L 2 -normalized solutions of the Gross- Pitaevskii (GP) equation with ring-shaped potential, which arises in Bose-Einstein condensate with attractive interaction. By establishing some delicate energy estimates, we prove that symmetry breaking occurs for the solution of GP equation as the interaction strength a > 0 approaches a critical value a, each L 2 -normalized solution of the GP equation concentrates to a point on the circular bottom of the potential well and then is non-radially symmetric as a a. However, when a > 0 is suitably small we prove that the L 2 -normalized solutions are unique, and the unique solution is radially symmetric. 11