Plenary talks
T. Bartsch, Gießen, Multi-bubble nodal solutions for an almost critical elliptic equation
Joint work with Teresa D'Aprile and Angela Pistoia.
We report on recent work on the existence and shape of nodal solutions of the problem \[ -\Delta u=|u|^{2^*-2-\epsilon}u \,\hbox{ in }\Omega, \quad u=0 \,\hbox{ on }\partial \Omega. \] Here \(\Omega\) is a smooth bounded domain in \(\mathbb{R}^N\), \(N\geq 3\), \(2^*=\frac{2N}{N-2}\), and \(\epsilon>0\) is a small parameter.
S. Cingolani, Bari, Morse index estimates for \(p\)-Laplace equations via uniform Sobolev inequalities
Joint work with M. Degiovanni and B. Sciunzi.
Let us consider the problem \[ (P) \ \ \left\{ \begin{array}{ll} - \Delta _{p}u = f(x,u) & \hbox{in} \ \Omega \\ \qquad u=0 & \hbox{on} \ \partial \Omega \end{array} \right. \] where \(\Omega\) is a bounded domain in \(\mathbb{R}^N\) with \(C^{1, \alpha}\) boundary, \(\Delta_p u= \mathrm{ div} (| \nabla u|^{p-2} \nabla u)\), \(p>2\), and \(f : \overline \Omega \times \mathbb{R} \to \mathbb{R}\) satisfies the following assumption: \(f \in C^1(\overline \Omega \times \mathbb{R})\) and for any \((x,s) \in \overline \Omega \times \mathbb{R}\), \(|D_s f (x,t)| \leq c_1 |t|^{q} + c_2\) with \(c_1,c_2\) positive constants and \(0 \leq q \leq p^*-2\) if \(N > p\), \(p^*=Np/(N-p)\) while \(q\) is any non negative number, if \(N=p\).
I will present some recent developments on Morse index estimates for each weak nontrivial solution \(u_0 \in W^{1,p}_0(\Omega)\) of the problem \((P)\) under a condition of totally boundedness. We will show that such a condition is, for instance, fulfilled if the \(f(x,s)>0\) for any \(x\in \overline \Omega\), \(s >0\) and \(f(x, s)=0\) for any \(x \in \overline \Omega\), \(s \leq 0\) or if \(\Omega\) is the ball or the annulus and \(f\) is changing sign function.
L. Dupaigne, Picardie, Monotonicity formulae and applications
I will present some recent results on regularity theory and Liouville-type theorems in the context of supercritical elliptic problems. I will highlight two beautiful classical ideas, originally introduced for the study of minimal surfaces : Fleming's blow-down method and Federer's dimension reduction principle.
B. Kawohl, Köln, On the longest shortest fence
Joint work with L. Esposito, V. Ferone, C. Nitsch and C. Trombetti.
Over 50 years ago Polya stated the following problem. Given a plane convex set \(K\) (a piece of land), find the shortest curve (or fence) that bisects this set into two subsets of equal area. Is it true that this curve is never longer than the diameter of the circular disc of same area as K? Under the additional assumption that K is centrosymmetric (i.e., \(K=-K\)) he gave a simple proof that this is indeed the case. Without this assumption the question is much harder to answer positively. A result of N. Fusco and A. Pratelli states, that if the fences are restricted to be straight line segments and if the sets are supposed to be convex Zindler sets, the answer is negative. In that case the longest shortest fence is attained for the Auerbach triangle and not for the disc. My coauthors and I were also able to show that the word Zindler can be removed from this statement.
A. Szulkin, Stockholm, On a quasilinear Schrödinger equation
Joint work with Xiangdong Fang.
We consider the semilinear Schrödinger equation \[ -\Delta u+V(x)u-\Delta(u^{2})u=g(x,u), \quad x\in \mathbb{R}^{N}, \] where \(g\) and \(V\) are periodic in \(x_1,\ldots,x_N\), \(V>0\), \(g\) is odd in \(u\) and of subcritical growth in the sense that \(|g(x,u)|\leq a(1+|u|^{p-1})\), where \(4<p < 2\cdot 2^*\). We show that this equation has infinitely many geometrically distinct solutions in each of the following two cases:
- \(g(x,u)=o(u)\) as \(u\to 0\), \(G(x,u)/u^4\to\infty\) as \(|u|\to\infty\), where \(G\) is the primitive of \(g\), and \(u\mapsto g(x,u)/u^3\) is positive for \(u\ne 0\), nonincreasing on \((-\infty, 0)\) and nondecreasing on \((0, \infty)\),
- \(g(x,u)=q(x)u^3\), where \(q>0\).
The argument uses the Nehari manifold technique. A special feature here is that the Nehari manifold is not likely to be of class \(C^1\).
S. Terracini, Torino, Uniform Hölder bounds for strongly competing systems involving standard and anomalous diffusions
Joint work with G. Verzini and A. Zilio.
For a class of competition-diffusion nonlinear systems involving \(s\)-Laplacian, including the fractional Gross-Pitaevskii system \[ (-\Delta)^{s} u_i=\omega_i u_i^3 + \lambda_i u_i -\beta u_i\sum_{j\neq i}a_{ij}u_j^2,\qquad i=1,\dots,k, \] we prove that \(L^\infty\) boundedness implies \(\mathcal{C}^{0,\alpha}\) boundedness for every \(\alpha\in[0,1/2)\) uniformly as \(\beta\to +\infty\). Moreover we prove that the limiting profile is \(\mathcal{C}^{0,1/2}\). This system arises, for instance, in the relativistic Hartree-Fock approximation theory for \(k\)-mixtures of Bose-Einstein condensates in different hyperfine states.
Wang Zhi Qiang, Utah State, Synchronized and segregated vector solutions for coupled nonlinear Schrodinger equations
Joint work with Shuangjie Peng.
We report recent work on new type of solutions for the coupled nonlinear Schrodinger equations. Depending upon the system being attractive or repulsive multiple (infinitely many) synchronized or segregated type solutions can be constructed, further making correlation between the coupling constants and the existence of different type of solutions.
T. Weth, Frankfurt, Real weakly decaying solutions of the nonlinear Helmholtz equation via variational methods
Joint work with G. Evéquoz.
I will discuss the existence and multiplicity of real solutions of the nonlinear Helmholtz equation \[ - \Delta u - k^2 u = f(x,u),\qquad x\in {\mathbb{R}}^N \] which satisfy the asymptotic conditions \[ u(x)=O(|x|^{\frac{1-N}{2}}) \quad \text{and} \quad |x|^{\frac{N-1}{2}}\left(\frac{\partial^2 u}{\partial r^2}(x)+k^2 u(x)\right) \to 0 \qquad \text{as $r=|x| \to \infty$.} \] Here \(f\) is a superlinear and subcritical nonlinearity. We develop different variational frameworks for this problem depending on the spatial asymptotics of \(f\). The main difficulty is given by the fact that zero is contained in the interior of the essential spectrum of the Schrödinger operator \(-\Delta -k^2\). Our results give rise to the existence of standing wave solutions of corresponding nonlinear Klein-Gordon equations with arbitrarily large frequency.
Contributed talks
Cyril J. Batkam, Université de Sherbrooke, Generalized fountain theorems and applications
The fountain theorems of Thomas Bartsch and Michel Willem, and their variants due to Wenming Zou, are effective tools of finding large and small energy solutions of some differential equations. By using the degree theory and the \(\tau\)-topology of Kryszewski and Szulkin, we establish some generalizations of these theorems to the case of differential equations which lead to strongly indefinite functionals. The new fountain theorems will be applied to some elliptic systems.
Isabel Coelho, Université libre de Bruxelles, Travelling wave profiles in some models with nonlinear diffusion
Joint work with Luís Sanchez and Carlos Fernández.
The results presented are versions of some classical results on the speed of travelling waves for FKPP equations considering different models of nonlinear diffusion. More precisely, we discuss existence and some properties of the monotone solutions of the boundary value problem \[ \left\{ \begin{aligned} (P (u'))' − cu '+ f (u) &= 0,\\ u(−\infty) &= 0,\\ u(+\infty) &= 1, \end{aligned} \right. \] which is motivated by the model case with \(P (x) = x/\sqrt{1 - x^2}\). This in turn corresponds to the problem of finding travelling waves for analogues of the FKPP equation. The parameter \(c\) is the wave speed and we assume that \(f\) is a continuous function in \([0, 1]\) such that \(f (0) = f (1) = 0\). We obtain a theory of admissible speeds and some other properties that generalise classical and recent results.
Jarosław Mederski, Toruñ Ground state solutions to a nonlinear Schrödinger equation with periodic potential
We study the following nonlinear Schrödinger equation \[ \left\{ \begin{aligned} -\Delta u + V(x) u &= g(x,u) & & \text{for \(x\in\mathbb{R}^N\)},\\ u(x) &\to 0 && \text{as } |x|\to\infty \end{aligned} \right. \] where \(V:\mathbb{R}^N\to\mathbb{R}\) and \(g:\mathbb{R}^N\times\mathbb{R}\to\mathbb{R}\) are periodic. We assume that \(0\) is a boundary point of the essential spectrum of \(-\Delta+V\). The problem has been studied by means of linking-type results and the existence of weak solutions has been proved. We present a Nehari manifold technique which allows to deal with the underlying geometry of the energy functional associated to the equation. The superlinear term \(g\) satisfies Nehari type monotone conditions and the existence of ground state solutions is obtained.
Manon Nys, Université libre de Bruxelles, On the eigenvalues of Aharonov-Bohm operators with varying poles
Joint work with Benedetta Noris and Susanna Terracini
We consider the following system \[ \left\{ \begin{aligned} (i \nabla + A_a)^2 \varphi_a &= \lambda_a \varphi_a & &\text{in \(\Omega\)},\\ \varphi_a &= 0 & & \text{on \(\partial \Omega\)}, \end{aligned} \right. \] where \(\Omega\) is a bounded domain of \(\mathbb{R}^2\) and \(A_a\) is the Aharonov-Bohm vector potential. It corresponds to the presence of a singular magnetic field at the point \(a \in \Omega\). We are mostly interested to study the behavior of the eigenvalues of the magnetic operator when the point \(a\) moves inside \(\Omega\) or to the boundary \(\partial \Omega\).
Hichem Ounaies, Monastir, Infinitely many solutions for a class of sublinear Schrödinger equations with sign-changing potentials
We study the following Schrödinger equation \[ -\Delta u + V(x) u = a(x)g(u) \qquad x \in \mathbb{R}^N, \quad N \ge 3, \] where \(V\) and \(a\) are functions with sign-changing on \(\mathbb{R}^N\) while the nonlinearity \(g : \mathbb{R} \to \mathbb{R}\) is sub-linear. Three cases will be considered about \(a\) and \(V\). In each situation, we prove the existence of infinitely many solutions.
Eugénio M. Rocha, Aveiro, Multiplicity results for some classes of Schrödinger--Poisson systems
Joint work with L. Huang and J. Chen.
We consider the Schrödinger-Poisson system \[ \left\{ \begin{array}{ll} -\Delta u +u +l(x)\phi u = f(x,u) \quad & \ \text{in}\ \mathbb{R}^3,\\ -\Delta \phi = l(x)u^2\quad & \ \text{in } \mathbb{R}^3, \end{array} \right. \] for suitable parameters. This system is obtained while looking for the existence of standing waves for the nonlinear Schrödinger equation interacting with an unknown electrostatic field. In this context the nonlinear term \(f\) simulates, as usual, the interaction between many particles, while the solution \(\phi\) of the Poisson equation plays the role of a potential determined by the charge of the wave function itself.
The system can be transformed into a nonlocal elliptic equation, for which in some situations we can apply suitable variational methods or concentration compactness techniques, but for other classes our methods neither use the Palais-Smale sequence nor Ekeland variational principle. In particular:
- When \((x,u)= k(x)|u|^{2^\ast-2}u +\mu h(x)u\) where \(\mu\) is a positive constant and the nonlinear growth reaches the Sobolev critical exponent, we prove the existence of at least a pair of fixed sign and a pair of sign-changing solutions in \(H^1(\mathbb{R}^3)\times D^{1, 2}(\mathbb{R}^3)\) under some suitable conditions on the non-negative functions \(l, k, h\), but not requiring any symmetry property on them.
- When \(f(x,u)= k(x)|u|^{p-2}u + h(x)u\) where \(4 < p < 6\), \(k\in C(\mathbb{R}^3)\), \(k\) changes sign in \(\mathbb{R}^3\) and \(\lim_{|x|\rightarrow\infty} k(x) = k_\infty < 0\), we mainly prove the existence of at least two positive solutions in the case that \(\mu > \mu_1\) and near \(\mu_1\), where \(\mu_1\) is the first eigenvalue of \(-\Delta + id\) in \(H^1(\mathbb{R}^3)\) with weight function h, whose corresponding eigenfunction is denoted by $e_1$. An interesting phenomena is that we do not need the condition \(\int_{\mathbb{R}^3} k(x)e^p_1\, dx < 0\), which has been shown to be a necessary condition to the existence of positive solutions for semilinear elliptic equations with indenite nonlinearity (e.g. see e.g. Alama et. al. Calc. Var. PDE 1 (1993), 439).
On going research, regarding a generalization of the system and possible relations with Superconductors, may be discussed if time allows it.
Alberto Saldana, Universität Frankfurt, Asymptotic symmetry of solutions of nonlinear competition-diffusion systems.
Joint work with Tobias Weth.
We consider a (parabolic) nonlinear competition-diffusion system under some symmetry assumptions, that is, in a bounded radial domain, and with non-autonomous radial nonlinearities. In this setting, we explore under which conditions all the solutions of the system exhibit an asymptotic (in time) symmetry. In this general framework, radial symmetry can not be expected, we look instead for foliated Schwarz symmetry, i.e., axial symmetry with some monotonicity properties. Some consequences for the elliptic version of the system are derived.
