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European Mathematical Society Publishing House
2024-03-28 09:54:24
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https://www.ems-ph.org/meta/jmeta-stream.php?jrn=JEMS&vol=8&iss=2&update_since=2024-03-28
Journal of the European Mathematical Society
J. Eur. Math. Soc.
JEMS
1435-9855
1435-9863
General
10.4171/JEMS
http://www.ems-ph.org/doi/10.4171/JEMS
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European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society
8
2006
2
Preface
General
155
156
10.4171/JEMS/42
http://www.ems-ph.org/doi/10.4171/JEMS/42
A note on a critical problem with natural growth in the gradient
Boumediene
Abdellaoui
Université Aboubekr Belkaïd, TLEMCEN, ALGERIA
Ireneo
Peral
Universidad Autónoma de Madrid, MADRID, SPAIN
Elliptic equations, Hardy potential, quadratic growth in the gradient, optimal summability
The paper analyzes the influence on the meaning of natural growth in the gradient, of a perturbation by a Hardy potential in some elliptic equations. We obtain a linear differential operator that, in a natural way, is the corresponding gradient for the perturbed elliptic problem. The main results are: i) Optimal summability of the data to have weak solutions; ii) Optimal linear operator associated, and, iii) Multiplicity and characterization of all solutions in terms of some measures. The results also are new for the Laplace operator perturbed for an inverse-square potential.
Partial differential equations
Functional analysis
General
157
170
10.4171/JEMS/43
http://www.ems-ph.org/doi/10.4171/JEMS/43
Concentration phenomena for Liouville's equation in dimension four
Adimurthi
Tata Institute of Fundamental Research, BANGALORE, INDIA
Frédéric
Robert
Université de Nice, NICE CEDEX 2, FRANCE
Michael
Struwe
ETH Zürich, ZÜRICH, SWITZERLAND
Calculus of variations and optimal control; optimization
General
171
180
10.4171/JEMS/44
http://www.ems-ph.org/doi/10.4171/JEMS/44
On two problems studied by A. Ambrosetti
David
Arcoya
Universidad de Granada, GRANADA, SPAIN
José
Carmona
Universidad de Almeria, ALMERIA, SPAIN
Ambrosetti-Prodi and Ambrosetti-Rabinowitz problems, topological degree, a priori bound
We study the Ambrosetti-Prodi and Ambrosetti-Rabinowitz problems. We prove for the first one the existence of a continuum of solutions with shape of a reflected C ($\supset$-shape). Next, we show that there is a relationship between these two problems.
Partial differential equations
Global analysis, analysis on manifolds
General
181
188
10.4171/JEMS/45
http://www.ems-ph.org/doi/10.4171/JEMS/45
A remark on the bifurcation diagrams of superlinear elliptic equations
Abbas
Bahri
Rutgers University, PISCATAWAY, UNITED STATES
We prove a formula relating the index of a solution and the rotation number of a certain complex vector along bifurcation diagrams.
Partial differential equations
General
189
193
10.4171/JEMS/46
http://www.ems-ph.org/doi/10.4171/JEMS/46
On the principal eigenvalue of elliptic operators in $\R^N$ and applications
Henry
Berestycki
Ecole des hautes études en sciences sociales, PARIS Cedex 13, FRANCE
Luca
Rossi
Università di Roma La Sapienza, ROMA, ITALY
Elliptic operators, principal eigenvalue, generalized principal eigenvalue in $\R^N$, limit periodic operators
Two generalizations of the notion of principal eigenvalue for elliptic operators in $\R^N$ are examined in this paper. We prove several results comparing these two eigenvalues in various settings: general operators in dimension one; self-adjoint operators; and ``limit periodic'' operators. These results apply to questions of existence and uniqueness for some semi-linear problems in all of space. We also indicate several outstanding open problems and formulate some conjectures.
Partial differential equations
General
195
215
10.4171/JEMS/47
http://www.ems-ph.org/doi/10.4171/JEMS/47
Spherical semiclassical states of a critical frequency for Schrödinger equations with decaying potentials
Jaeyoung
Byeon
KAIST, DAEJEON, SOUTH KOREA
Zhi-Qiang
Wang
Utah State University, LOGAN, UNITED STATES
Nonlinear Schrödinger Equations, critical frequency, concentrations on spheres
For singularly perturbed Schr\"odinger equations with decaying potentials at infinity we construct semiclassical states of a critical frequency concentrating on spheres near zeroes of the potentials. The results generalize some recent work of Ambrosetti-Malchiodi-Ni[3] which gives solutions concentrating on spheres where the potential is positive. The solutions we obtain exhibit different behaviors from the ones given in [3].
Partial differential equations
General
217
228
10.4171/JEMS/48
http://www.ems-ph.org/doi/10.4171/JEMS/48
On bifurcation and uniqueness results for some semilinear elliptic equations involving a singular potential
Manuela
Chaves
Universidad Autónoma de Madrid, MADRID, SPAIN
Jesús
García-Azorero
Universidad Autónoma de Madrid, MADRID, SPAIN
We will present some results concerning the problem \begin{equation}\label{eq:sl} \left\{ \begin{array}{ll} & -\D u = \lambda \dfrac {u}{|x|^2}+u^q \, , \, \, u > 0 \hbox{ in }\O,\\ & u|_{\p \O}=0, \end{array} \right. \end{equation} where $0
Partial differential equations
General
229
242
10.4171/JEMS/49
http://www.ems-ph.org/doi/10.4171/JEMS/49
General results on the eigenvalues of operators with gaps, arising from both ends of the gaps. Application to Dirac operators
Jean
Dolbeault
Université de Paris Dauphine, PARIS CEDEX 16, FRANCE
Maria
Esteban
Université de Paris Dauphine, PARIS CEDEX 16, FRANCE
Eric
Séré
Université de Paris Dauphine, PARIS CEDEX 16, FRANCE
This paper is concerned with an extension and reinterpretation of previous results on the variational characterization of eigenvalues in gaps of the essential spectrum of self-adjoint operators. We state two general abstract results on the existence of eigenvalues in the gap and a continuation principle. Then these results are applied to Dirac operators in order to characterize simultaneously eigenvalues corresponding to electronic and positronic bound states.
Operator theory
Quantum theory
General
243
251
10.4171/JEMS/50
http://www.ems-ph.org/doi/10.4171/JEMS/50
On the number of positive solutions of singularly perturbed 1D NLS
Patricio
Felmer
Universidad de Chile, SANTIAGO, CHILE
Salomé
Martínez
Universidad de Chile, SANTIAGO, CHILE
Kazunaga
Tanaka
Waseda University, TOKYO, JAPAN
Nonlinear Schrödinger equations, singular perturbations, adiabatic profiles
We study singularly perturbed 1D nonlinear Schr\"odinger equations (\ref{eq:1.1}). When $V(x)$ has multiple critical points, (\ref{eq:1.1}) has a wide variety of positive solutions for small $\varepsilon$ and the number of positive solutions increases to $\infty$ as $\varepsilon\to 0$. We give an estimate of the number of positive solutions whose growth order depends on the number of local maxima of $V(x)$. Envelope functions or equivalently adiabatic profiles of high frequency solutions play an important role in the proof.
Partial differential equations
General
253
268
10.4171/JEMS/51
http://www.ems-ph.org/doi/10.4171/JEMS/51
Multiplicity results for a family of semilinear elliptic problems under local superlinearity and sublinearity
Djairo
de Figueiredo
IMECC - UNICAMP, CAMPINAS, BRAZIL
Jean-Pierre
Gossez
Université Libre de Bruxelles, BRUXELLES, BELGIUM
Pedro
Ubilla
Universidad de Santiago de Chile, SANTIAGO, CHILE
Multiplicity, semilinear elliptic problem, local sub and superlinear nonlinearities, concave-convex nonlinearities, critical exponent, upper and lower solutions, variational method
In this paper we study the existence, nonexistence and multiplicity of positive solutions for the family of problems $ -\Delta u = f_\lambda (x,u)$, $u \in H^1_0(\Omega)$, where $\Omega$ is a bounded domain in $\mathbb{R}^N$, $N\geq 3 $ and $\lambda>0$ is a parameter. The results include the well-known nonlinearities of the Ambrosetti-Brezis-Cerami type in a more general form, namely $ \lambda a (x)u^q + b(x) u^p $, where $0 \leq q
Partial differential equations
Global analysis, analysis on manifolds
General
269
288
10.4171/JEMS/52
http://www.ems-ph.org/doi/10.4171/JEMS/52
Sharp estimates for the Ambrosetti-Hess problem and consequences
José
Gámez
Universidad de Granada, GRANADA, SPAIN
Juan
Ruiz-Hidalgo
I.E.S. Antonio de Mendoza, ALCALÁ LA REAL, JAÉN, SPAIN
Motivated by {3}, we define the ``Ambrosetti-Hess problem'' to be the problem of bifurcation from infinity and study of the local behavior of continua of solutions of nonlinear elliptic eigenvalue problems. Although the works in this way underline the asymptotic properties on the nonlinearity, here we point out that this local behavior is determined by the global shape of the nonlinearity.
Partial differential equations
General
287
294
10.4171/JEMS/53
http://www.ems-ph.org/doi/10.4171/JEMS/53
A fully nonlinear version of the Yamabe problem on manifolds with boundary
YanYan
Li
Rutgers University, PISCATAWAY, UNITED STATES
Aobing
Li
University of Wisconsin-Madison, MADISON CITY, UNITED STATES
We propose a fully nonlinear version of the Yamabe problem on manifolds with boundary. The boundary condition for the conformal metric is the mean curvature. We establish some Liouville type theorems and Harnack type inequalities.
Manifolds and cell complexes
General
295
316
10.4171/JEMS/54
http://www.ems-ph.org/doi/10.4171/JEMS/54
A geometric problem and the Hopf Lemma. I
YanYan
Li
Rutgers University, PISCATAWAY, UNITED STATES
Louis
Nirenberg
New York University, NEW YORK, UNITED STATES
A classical result of A.~D. Alexandrov states that a connected compact smooth $n$-dimen\-sional manifold without boundary, embedded in $ {\mathbb R}^{n+1}$, and such that its mean curvature is constant, is a sphere. Here we study the problem of symmetry of $M$ in a hyperplane $X_{n+1}={\rm const}$ in case $M$ satisfies: for any two points $(X', X_{n+1})$, $(X', \widehat X_{n+1})$ on $M$, with $X_{n+1}>\widehat X_{n+1}$, the mean curvature at the first is not greater than that at the second. Symmetry need not always hold, but in this paper, we establish it under some additional condition for $n=1$. Some variations of the Hopf Lemma are also presented. Part II [Y.Y. Li and L. Nirenberg, Chinese Ann. Math. Ser. B 27 (2006), 193--218] deals with corresponding higher dimensional problems. Several open problems for higher dimensions are described in this paper as well.
Partial differential equations
General
317
339
10.4171/JEMS/55
http://www.ems-ph.org/doi/10.4171/JEMS/55
Multiplicity and stability of closed geodesics on Finsler 2-spheres
Yiming
Long
NANKAI UNIVERSITY, TIANJIN, CHINA
Closed geodesics, Finsler metric, 2-sphere
A survey of recent progress on the multiplicity and stability problems of closed geodesics on Finsler $2$-spheres is given.
Differential geometry
Global analysis, analysis on manifolds
General
341
353
10.4171/JEMS/56
http://www.ems-ph.org/doi/10.4171/JEMS/56
On the necessity of gaps
Hiroshi
Matano
University of Tokyo, TOKYO, JAPAN
Paul
Rabinowitz
University of Wisconsin, MADISON, UNITED STATES
Recent papers have studied the existence of phase transition solutions for model equations of Allen-Cahn type equations. These solutions are either single or multi-transition spatially heteroclinics or homoclinics between simpler equilibrium states. A sufficient condition of the construction of the multi-transition solutions is that there are gaps in the ordered set of single transition solutions. In this paper we explore the necessity of these gap conditions.
Partial differential equations
General
355
373
10.4171/JEMS/57
http://www.ems-ph.org/doi/10.4171/JEMS/57
The periodic Ambrosetti-Prodi problem for nonlinear perturbations of the p-Laplacian
Jean
Mawhin
Université Catholique de Louvain, LOUVAIN-LA-NEUVE, BELGIUM
Ambrosetti-Prodi problem, periodic solutions, upper and lower solutions, topological degree
We prove an Ambrosetti-Prodi-type result for the periodic solutions of equation $\left(|u'|^{p-2}u')\right)' + f(u)u' + g(x,u) = t ,$ when $f$ is arbitrary and $g(x,u) \to +\infty$ or $g(x,u)\to - \infty$ when $|u| \to \infty.$ The proof uses upper and lower solutions and Leray-Schauder degree.
Partial differential equations
General
375
388
10.4171/JEMS/58
http://www.ems-ph.org/doi/10.4171/JEMS/58
A maximum principle for mean-curvature type elliptic inequalities
James
Serrin
University of Minnesota, MINNEAPOLIS, UNITED STATES
Quasilinear elliptic inequalities, maximum principles, mean curvature equation
Consider the divergence structure elliptic inequality $$ {\rm div}\{\boldsymbol A(x,u,Du)\} + B(x,u,Du) \ge 0 \leqno (1) $$ in a bounded domain $\Omega\subset \RR^n$. Here $$ \boldsymbol A(x,z,\boldsymbol \xi): \,K \to \RR^n; \qquad B(x,z,\boldsymbol \xi): \,K \to \RR, \qquad K = \Omega \times \RR^+ \times \RR^n, $$ and $\boldsymbol A$, $B$ satisfy the following conditions $$ \begin{aligned} \langle\boldsymbol A(\boldsymbol \xi) , \boldsymbol \xi \rangle \ge |\xi| - & c(x)z - a(x), \qquad |\boldsymbol A(x,z,\boldsymbol \xi)| \le \mbox{Const.},\\ & B(x,z,\boldsymbol \xi) \le b(x),\end{aligned} $$ for all $(x,z,\boldsymbol \xi)) \in K$, where $a(x)$, $b(x),\ c(x)$ are given non-negative functions. Our interest is in the validity of the maximum principle for solutions of (1), that is, the statement that {\it any solution which satisfies $u\le 0$ on $\partial\Omega$ must be a priori bounded above in $\Omega$.} This question arises, in particular, when one is interested in the mean curvature equation $$ {\rm div}\frac{Du}{\sqrt{1+|Du|^2}} = nH(x). $$
Partial differential equations
General
389
398
10.4171/JEMS/59
http://www.ems-ph.org/doi/10.4171/JEMS/59
Uniqueness and stability of ground states for some nonlinear Schrödinger equations
Charles
Stuart
Ecole Polytechnique Federale, LAUSANNE, SWITZERLAND
Orbital stability, standing waves, ground states
We discuss the orbital stability of standing waves of a class of nonlinear Schrödinger equations in one space dimension. The crucial feature for our treatment is the presence of a non-constant linear potential that is even and decreasing away from the origin in space. This enables us to establish the orbital stability of all ground states over the whole range of frequencies for which such solutions exist.
Partial differential equations
Dynamical systems and ergodic theory
General
399
414
10.4171/JEMS/60
http://www.ems-ph.org/doi/10.4171/JEMS/60