A note on the problem −Δu=λu+u|u|2*−2

The dual approach to the problem –u=u+u|u|^2*–2, uþe ₀ ¹ (gW), permits a simple proof of a recent existence result [5] and allows extensions of this result to similar problems also with asymmetric mnonlinearities.Supported by min P.I., Gruppo Naz. (40%) Calcolo delle Variazioni...     

On the equation −Δu + e u − 1 = 0 with measures as boundary data

If Ω is a bounded domain in ${\mathbb{R}^N}$ , we study conditions on a Radon measure μ on ?Ω for solving the equation ?Δu + e ^u ? 1 = 0 in Ω with u = μ on ?Ω. The conditions are expressed in terms of Orlicz capacities.     

On the Greatest Common Di visor of u − 1 and v − 1 with u and v Near ${\cal S}$- units

In [2], it was shown that if a and b are multiplicatively independent integers and ɛ > 0, then the inequality gcd (aⁿ − 1,bⁿ − 1) < exp(ɛn) holds for all but finitely many positive integers n. Here, we generalize the above result. In particular, we show that if f(x),f₁(x),g(x),g₁(x) are non-zero polynomials with integer coefficients, then for every ɛ > 0, the inequality holds for all but finitely many positive integers n.     

The behavior of bounded solutions to the equation Δu−c(x)u=0 on riemannian manifolds of special type

In the paper we consider solutions of the equation Δu−c(x)u=0,c(x)≥0, on complete Riemannian manifolds constituted as follows: the exterior of some compact set is isometric to the direct product of the semiaxis by some compact manifold with the metricds ²=h ²(r)dr ²+g ²(r)dθ². Necessary and sufficient conditions under which bounded solutions of the equation have a limit independent of θ asr→∞ are obtained and also conditions under which the two-sided Liouville theorem is valid. Translated fromMatematicheskie Zametki, Vol. 65, No. 2, pp. 215–221, February, 1999.     

Regularity of ∫Ω|∇u|2 + λ∫Ω|u−f{2 and some gap phenomenon

We study the regularity of the minimizer u for the functional F (u,f)=|u|² + |u–f{² over all maps uH ¹(, S ²). We prove that for some suitable functions f every minimizer u is smooth in if ₀ and for the same functions f, u has singularities when is large enough.

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Résumé On étudie la régularité des minimiseurs u du problème de minimisation min_ueH ¹(,S²)(|u|² + |u–f{². On montre que pour certaines fonctions f, u est régulière lorsque ₀ et pour les mêmes f, si est assez grand, alors u possède des singularités.

     

Remarks on bifurcations ofu + μu − u k=0 (4≤kεZ +)

In this paper, we investigate the bifurcations of one class of steady-state reaction-diffusion equations of the formu″ + μu − u ^k=0, subjectu(0)=u(π)=0, where μ is a parameter, 4≤kεZ ⁺. Using the singularity theory based on the Liapunov-Schmidt reduction, some satisfactory results are obtained. This work is supported by the National Natural Science Foundation of China (No.19971057) and the Youth Science Foundation of Shanghai Municipal Commission of Education (No.99QA66).     

On radially symmetric solutions of the equation ?Δu + u = |u| p-1 u. An ODE approach

Questions of the existence in a ball of radially symmetric solutions of the equation indicated in the title with the Dirichlet zero boundary conditions are studied in many publications and generally speaking, there was obtained more or less complete answer on these questions. It is known now that if the dimension of the space d????3 and 1 <?p?<?(d?+ 2)/(d ? 2) or if d?=?2 and p?>?1, then for any integer l??? 0 this problem in a ball or in the entire space ${x \in \mathbb {R}^d}$ has a radially symmetric solution with precisely l zeros as a function of r?=?|x|. If d??? 3 and p????(d?+?2)/(d ? 2), then the problem in the entire space has no nontrivial solution. For the first time, this problem was studied by a variant of the variational method. However, it is known to the specialists in the field that it is also interesting to obtain the same results by using methods of the qualitative theory of ODEs. In the present article, we shall give a simple proof of the result above in this way. An earlier proof of this result of the other authors is essentially more complicated than our one.     

Smoothness and asymptotics of global positive branches of Δu+λf(u)=0

The smallest eigenvalue of v+ f(0)v=0 gives rise to a global positive branch of u+ f(u)=0 in ² together with homogeneous Dirichlet boundary conditions (f(0)=0,f(0)0). The theory, however, guarantees only that this branch is a continuum. We present a result that in a reasonably large class of problems this branch is actually an unbounded smooth curve. In the second part we study the asymptotic behavior of this positive smooth curve whenf has exponential growth. All results apply also to branches emanating at higher eigenvalues. As shown in [7] they have a fixed nodal structure when restricted to appropriate fixed-point spaces.Dedicated to Prof. Klaus Kirchgässner on the occasion of his sixtieth birthday