Abstracts

Topological and variational methods for the Moser-Trudinger equation
Luca Martinazzi (University of Padova)
We discuss the existence of critical points of the Moser-Trudinger functional in dimension 2
with arbitrarily prescribed Dirichlet energy using both degree theory (Leray-Schauder degree)
and variational methods (baricenter technique of Djadli-Malchiodi and Struwe monotonicity).
This talk is based on joint works with Francesca De Marchis, Olivier Druet, Andrea Malchiodi,
Gabriele Mancini and Pierre-Damien Thizy.

On the proﬁle decomposition of a volume functional
Michinori Ishiwata (Osaka University)
The existence of a constant mean curvature surface can be treated by the variational
method. To obtain the existence of the surface, it is important to analyse the nature of thebreakdown of the compactness property of the corresponding functional which is done by
the celebrated papers of Brezis-Coron. In this talk, we shed some light on this issue fromthe proﬁle-decomposition concerning the Sobolev embedding into BMO together with thecompensated-compactness which will give an improved version of Wente’ s inequality.

Convex properties of positive solutions for a class of quasi-linear
elliptic problems
Tatsuya Watanabe (Kyoto Sangyo University)

We are interested in the convexity of superlevel sets of positive solutions for a class ofquasilinear elliptic equations in bounded strictly convex domains. Our ﬁrst result states thatany positive solution has convex superlevel sets for nonlinearities of sublinear type. We further
show that any positive semi-stable solution has convex superlevel sets. The existence ofpositive semi-stable solutions are also investigated for various types of nonlinear terms andquasilinear terms. This talk is based on a joint work with M. Squassina (Universita Cattolicadel Sacro Cuore).

The eﬀect of heterogeneity on one-peak stationary solutions to theSchnakenberg model
Yuta Ishii (Tokyo Metropolitan University)

In this talk, we consider the Schnakenberg model on the interval (−1, 1) with heterogeneityg(x) in front of the nonlinear term under the Neumann boundary condition. We ﬁrst construct
one-peak stationary solutions by using the Liapunov-Schmidt reduction method. Moreover,
we study the linear stability of the solutions above in details. In previous works, one-peaksymmetric solutions have been studied for the symmetric heterogeneity case including g(x) =1. In this work, since g(x) may be not symmetric on the interval (−1, 1), the constructed
solution may be not symmetric. In particular, we reveal the eﬀect of the heterogeneity on thelocation of the concentration point of the solution and the stability, which is a new phenomenon
compared with the symmetric heterogeneity case.

Asymptotic property of ground states for a class of quasilinear Schrdinger
equations with H 1 -critical growth
Masataka Shibata (Tokyo Institute of Technology)
We consider a quasilinear elliptic equation −∆u − κu∆(u2 ) = up − u in RN , where N ≥ 3,p > 1, κ > 0. The equation appears in the study of standing waves of a modiﬁed Schrödingerequation. It has no solution if p + 1 ≥ 2 · 2∗ and has a unique solution uκ ∈ H 1 (RN ) if
2 < p + 1 < 2 · 2∗ , where 2∗ = 2N/(N − 2) is the Sobolev critical exponent. We consider
the asymptotic behavior of the solution uκ as κ → 0. The asymptotic behavior depends on pdeeply. Indeed, in the case p + 1 < 2∗ (H 1 -subcritical case), uκ converges to a unique solutionof the scalar ﬁeld equation. In the case p + 1 > 2∗ (H 1 -supercritical case), uκ converges to aunique solution of some limiting problem under suitable scaling. In this talk, we discuss theasymptotic behavior in the case p + 1 = 2∗ (H 1 -critical case).

Variational construction of orbits realizing symbolic sequences in the planar
Sitnikov problem
Mitsuru Shibayama (Kyoto University)
Using the variational method, Chenciner and Montgomery (2000 Ann. Math. 152 881901) proved the existence of an eight-shaped orbit of the planar three-body problem withequal masses. Since then a number of solutions to the N-body problem have been discovered.
On the other hand, symbolic dynamics is one of the most useful methods for understanding
chaotic dynamics. The Sitnikov problem is a special case of the three-body problem. Thesystem is known to be chaotic and was studied by using symbolic dynamics (J.Moser, Stableand random motions in dynamical systems, Princeton University Press, 1973). In this talk, we
study the limiting case of the Sitnikov problem. By using the variational method, we show theexistence of various kinds of solutions in the planar Sitnikov problem. For a given symbolic
sequence, we show the existence of orbits realizing it. We also prove the existence of periodicorbits.

Strongly perturbed Moser-Trudinger functionals and their critical points indimension two.
Gabriele Mancini (Sapienza University of Rome)

I will discuss some recent results obtained in collaboration with P-D. Thizy concerningsemilinear elliptic problems involving Moser-Trudinger type nonlinearities. After summarizing
some known facts regarding the standard Moser-Trudinger critical equation, I will describe
how the classical results are aﬀected by the presence of perturbation terms. In particular, I will
present the answer to a question formulated by O. Druet, proving the existence of families ofnonlinearities with uniformly critical growth for which the corresponding semilinear problem
admits a bubbling sequence of solutions having a nontrivial weak limit

Eﬀect of compact term on maximization problem associated with
Trudinger-Moser inequality
Masato Hashizume (Ehime University)

It is well known that there exists a extremal for the original Trudinger-Moser inequality.
Recently, it was shown that perturbations have an eﬀect on existence and non-existence. In
this talk, we consider maximization problem associated with the Trudinger-Moser inequality
with lower order term. We obtain a threshold separating existence and non-existence.

The behavior of blow-up solutions for mean ﬁeld equations withprobability measure
Yohei Toyota (National Institute of Technology, Nara College)

In this talk we are concerned with several mean ﬁeld equations with probability measure,in particular,∫αeαv−∆v = λ ∫ αv P(dα) in Ω, v = 0 on ∂Ω,I Ω e dx
and∫αeαv P(dα)
eαv dxP(dα)Ω×I−∆v = λ ∫∫ Iin Ω,

v = 0 on ∂Ω,where Ω ⊂ R2 is a smooth bounded domain, λ > 0 is a constant and P(dα) is a Borel
probability measure on I = [0, 1]. The main topic in my talk is the family of unbounded
solutions for such mean ﬁeld equations. In terms of blow-up analysis, we shall show thebehavior of blow-up solutions near the blow-up points under the suitable assumptions of P.
This is partly joint work with professor T. Suzuki in Osaka University.

Solvability of a semilinear heat equation via a quasi scale invariance
Norisuke Ioku (Tohoku University)

Classiﬁcation theory on local in time solvability of nonlinear heat equations is investigated.
Without assuming a concrete growth rate on a nonlinear term, we reveal the threshold integrability of initial data which classify existence and nonexistence of solutions via a quasi-scaling
and its invariant integral. Global in time solvability is also studied. Typical nonlinear terms,
for instance polynomial type, exponential type and its sum, product and composition, can betreated as applications. This is a joint work with Yohei Fujishima(Shizuoka University).