On the Existence and the Profile of Nodal solutions of Elliptic Equations Involving Critical Growth

We study the existence of sign changing solutions to the slightly subcritical problem

Minimal nodal solutions of the pure critical exponent problem on a symmetric domain

We establish existence of nodal solutions to the pure critical exponent problem in u = 0 on where a bounded smooth domain which is invariant under an orthogonal involution of We extend previous results for positive solutions due to Coron, Dancer, Ding, and Passaseo to existence and multiplicity results for solutions which change sign exactly once.Received: 4 April 2003, Accepted: 26 August 2003, Published online: 24 November 2003Mathematics Subject Classification (2000): 35J65, 35J20Research partially supported by PAPIIT, UNAM, under grant IN110902-3.     

Multiple solutions of constant sign for nonlinear nonsmooth eigenvalue problems near resonance

In this paper we study a class of nonlinear elliptic eigenvalue problems driven by the p-Laplacian and having a nonsmooth locally Lipschitz potential. We show that as the parameter approaches (= the principal eigenvalue of ) from the right, the problem has three nontrivial solutions of constant sign. Our approach is variational based on the nonsmooth critical point theory for locally Lipschitz functions. In the process of the proof we also establish a generalization of a recent result of Brezis and Nirenberg for C₀¹ versus W₀^1,p minimizers of a locally Lipschitz functional. In addition we prove a result of independent interest on the existence of an additional critical point in the presence of a local minimizer of constant sign. Finally by restricting further the asymptotic behavior of the potential at infinity, we show that for all the problem has two solutions one strictly positive and the other strictly negative.Received: 7 January 2003, Accepted: 12 May 2003, Published online: 4 September 2003Mathematics Subject Classification (2000): 35J20, 35J85, 35R70     

Asymptotic behavior for elliptic problems with singular coefficient and nearly critical Sobolev growth

Let be a ball centered at the origin with radius R. We investigate the asymptotic behavior of positive solutions for the Dirichlet problem in on ∂B_R when ɛ→⁺ for suitable positive numbers μ Mathematics Subject Classification (2000) 35J60, 35B33     

Splitting lemma at infinity and a strongly resonant problem with periodic nonlinearity

In this paper we study the following problem:with periodic nonlinearity g, where and λ₂ is the second eigenvalue of −Δ, on H ¹ ₀(B). We proved that the problem has infinitely many solutions under some additional conditions on g and h. The method we used is a new variational reduction method. Mathematics Subject Classi cation (2000) 35J20, 35J70     

On the Existence and the Uniform Decay of a Hyperbolic Equation with Non-Linear Boundary Conditions

In this paper, we study a hyperbolic model based on the equation with nonlinear boundary conditions given by .We prove the existence and the uniqueness of global solutions. Also, we obtain the uniform decay of the energy without control of its derivative sign.AMS Subject Classification (2000), 35L05, 35L70, 35B40     

On the existence of multiple positive solutions of quasilinear elliptic eigenvalue problems

We study a quasilinear elliptic problem

Nonlinear elliptic equations with large supercritical exponents

The paper deals with the existence of positive solutions of the problem -Δ u=u^p in Ω, u=0 on ∂Ω, where Ω is a bounded domain of , n≥ 3, and p>2. We describe new concentration phenomena, which arise as p→ +∞ and can be exploited in order to construct, for p large enough, positive solutions that concentrate, as p→ +∞, near submanifolds of codimension 2. In this paper we consider, in particular, domains with axial symmetry and obtain positive solutions concentrating near (n-2)-dimensional spheres, which approach the boundary of Ω as p→ +∞. The existence and multiplicity results we state allow us to find positive solutions, for large p, also in domains which can be contractible and even arbitrarily close to starshaped domains (while no solution can exist if Ω is starshaped and , as a consequence of the Pohožaev's identity). Mathematics Subject Classification (2000) 35J20, 35J60, 35J65     

Liouville-type results for semilinear elliptic equations in unbounded domains

This paper is devoted to the study of some class of semilinear elliptic equations in the whole space:

Single-point condensation phenomena for a four-dimensional biharmonic Ren–Wei problem

We continue to study the asymptotic behavior of least energy solutions to the following fourth order elliptic problem (E _p ): as p gets large, where Ω is a smooth bounded domain in R ⁴ . In our earlier paper (Takahashi in Osaka J. Math., 2006), we have shown that the least energy solutions remain bounded uniformly in p and they have one or two “peaks” away form the boundary. In this note, following the arguments in Adimurthi and Grossi (Proc. AMS 132(4):1013–1019, 2003) and Lin and Wei (Comm. Pure Appl. Math. 56:784–809, 2003), we will obtain more sharper estimates of the upper bound of the least energy solutions and prove that the least energy solutions must develop single-point spiky pattern, under the assumption that the domain is convex.     

On nonlocal calculation for inhomogeneous indefinite Neumann boundary value problems

This paper concerns with a family of inhomogeneous Neumann boundary value problems having indefinite nonlinearities which depend on a real parameter . We discuss the existence and the multiplicity of positive solutions with respect to . Developing the fibering method further, we can introduce a constructive concept of the calculation of certain nonlocal intervals , the so-called sufficient intervals of the existence. Then we are able to prove some new results on the existence and the multiplicity of positive solutions for .Received: 22 December 2003, Accepted: 29 January 2004, Published online: 16 July 2004Mathematics Subject Classification (2000): 35J70, 35J65, 47H17     

Fourth order approximation of harmonic maps from surfaces

Let be a compact Riemannian surface and let be a compact Riemannian manifold, both without boundary, and assume that N is isometrically embedded into some ℝ ^l . We consider a sequence of critical points of the functional with uniformly bounded energy. We show that this sequence converges weakly in and strongly away from finitely many points to a smooth harmonic map. One can perform a blow-up to show that there separate at most finitely many non-trivial harmonic two-spheres at these finitely many points. Finally we prove the so called energy identity for this approximation in the case that ↪ ℝ ^l . Mathematics Subject Classification (2000) 58E20, 35J60, 53C43     

A geometric analysis of Renegar’s condition number,and its interplay with conic curvature

For a conic linear system of the form Ax ∈K, K a convex cone, several condition measures have been extensively studied in the last dozen years. Among these, Renegar’s condition number is arguably the most prominent for its relation to data perturbation, error bounds, problem geometry, and computational complexity of algorithms. Nonetheless, is a representation-dependent measure which is usually difficult to interpret and may lead to overly conservative bounds of computational complexity and/or geometric quantities associated with the set of feasible solutions. Herein we show that Renegar’s condition number is bounded from above and below by certain purely geometric quantities associated with A and K; furthermore our bounds highlight the role of the singular values of A and their relationship with the condition number. Moreover, by using the notion of conic curvature, we show how Renegar’s condition number can be used to provide both lower and upper bounds on the width of the set of feasible solutions. This complements the literature where only lower bounds have heretofore been developed.     

Non-variational elliptic systems in dimension two: a priori bounds and existence of positive solutions

We establish a priori bounds for positive solutions of semilinear elliptic systems of the form

Steady fronts solutions of semilinear elliptic equations in exterior domains arising in flame propagation

This article deals with a semilinear elliptic equation arising in a model for flame propagation outside a compact subset of , for n≥ 2. Our aim is to prove various kind of existence, nonexistence, uniqueness, monotonicity properties and singular limit results for the solutions T of the following Dirichlet problem in :

On the Lax Solution to the Hamilton–Jacobi Equation and Its Generalizations. Part II

This article is a continuation of [J. Math. Sci., 99, No.5, 1541–1547 (2000)] devoted to the validity of the Lax formula (cited in the article of Crandall, Ishii, and Lions [Bull. AMS, 27, No.1, 1–67 (2000)])

Harnack’s principle for quasiminimizers

Abstract We study Harnack type properties of quasiminimizers of the -Dirichlet integral on metric measure spaces equipped with a doubling measure and supporting a Poincaré inequality. We show that an increasing sequence of quasiminimizers converges locally uniformly to a quasiminimizer, provided the limit function is finite at some point, even if the quasiminimizing constant and the boundary values are allowed to vary in a bounded way. If the quasiminimizing constants converge to one, then the limit function is the unique minimizer of the -Dirichlet integral. In the Euclidean case with the Lebesgue measure we obtain convergence also in the Sobolev norm. Keywords: Metric space, doubling measure, Poincaré inequality, Newtonian space, Harnack inequality, Harnack convergence theorem Mathematics Subject Classification (2000): 49J52, 35J60, 49J27     

Quasiminima of the Lipschitz extension problem

In this paper, we extend the notion of quasiminimum to the framework of supremum functionals by studying the model case

On Galois groups of unramified pro-p extensions

Let p be an odd prime satisfying Vandiver’s conjecture. We consider two objects, the Galois group X of the maximal unramified abelian pro-p extension of the compositum of all Z _p -extensions of Q(μ _p ) and the Galois group of the maximal unramified pro-p extension of Q . We give a lower bound for the height of the annihilator of X as an Iwasawa module. Under some mild assumptions on Bernoulli numbers, we provide a necessary and sufficient condition for to be abelian. The bound and the condition in the two results are given in terms of special values of a cup product pairing on cyclotomic p-units. We obtain in particular that, for p  <  1,000, Greenberg’s conjecture that X is pseudo-null holds and is in fact abelian.