1,720,976 research outputs found
A multiplicity result for elliptic equations at critical growth in low dimension
No full textWe consider the problem -Deltau = \u\(2*-2)u + lambdau in Omega, u = 0 on partial derivativeOmega, where Omega is an open regular subset of R-N (N greater than or equal to 3), 2* = 2N/N - 2 is the critical Sobolev exponent and lambda is a constant in]0, lambda(1)[ where lambda(1) is the first eigenvalue of -Delta. In this paper we show that, when N greater than or equal to 4, the problem has at least N/2 + 1 (pairs of) solutions, improving a result obtained in [4] for N greater than or equal to 6
Infinitely many positive solutions to some nonsymmetric scalar field equations: the planar case
No full textWe show the existence of infinitely many positive solutions u ∈ H1(R2) to the
equation −Delta u + a(x)u = u^p, with p > 1 , without asking, on the positive potential a(x),
any symmetry assumption as inWei and Yan (Calc Var Partial Differ Equ 37, 423–439, 2010)
or Devillanova and Solimini (Adv Nonlinear Studies 12, 173–186, 2012) or small oscillation
assumption as in Cerami et al. (Commun Pure Appl Math, doi:10.1002/cpa.21410, 2012)
6 and in Weiwei and Wei (Infinitely many positive solutions for Nonlinear equations with
non-symmetric Potential, 2012)
A note on a resonance problem
No full textSynopsisIn this paper, we prove the existence of at least one solution to the problemwhere ∆k is an eigenvalue of the linear part, h is orthogonal to the eigenspace corresponding to ∆R and g is a nonlinear perturbation which can be, for instance, a continuous periodic real function with mean value zero. We employ the techniques used by the second author in a previous paper in which the same result was obtained in the case in which ∆R is assumed to be simple. The final result is obtained by using variational methods and in particular a suitable version of the saddle point theorem of P. Rabinowitz.</jats:p
Min-max levels transition in parametrized elliptic problems on unbounded domains
No full textWe discuss some attempts to apply the classical variational methods, in the spirit of Ambrosetti-Rabinowitz, to parametrized elliptic problems defined on unbounded domains. We construct some min-max classes and establish some estimates which allow, for instance, to conclude that, while for a generic coefficient there is always a solution for all small values of the parameter, in the case of a coefficient with a suitable exponential decay and in dimension N = 2 the set of the values of the parameter for which the problem has a nontrivial solution is unbounded
Some existence results for superlinear elliptic boundary value problems involving critical exponents
No full textAbstractThe existence of solutions to the problem −Δu − λu = u¦u¦2∗ − 2 in Ωu¦∂Ω = 0 is studied. For an arbitrary domain Ω ⊂Rn, if λ ϵ ]0, λ1[ and n ⩾ 6, the existence of solutions of changing sign is obtained. If Ω = BR(0) ⊂ Rn, λ ϵ ]0, λ1[, and n ⩾ 7, infinitely many radial solutions to this problem are exhibited, characterized by the number of nodes they possess
Tomography: mathematical aspects and applications
No full textIn this article we present a review of the Radon transform and the instability of the tomographic reconstruction process. We show some new mathematical results in tomography obtained by a variational formulation of the reconstruction problem based on the minimization of a Mumford–Shah type functional. Finally, we exhibit a physical interpretation of this new technique and discuss some possible generalizations
Regularity properties of free discontinuity sets
No full textThis paper is concerned with the problem of estimating the dimension, expected to be N - 2, of the singular set of a minimizer of a functional with free discontinuities in N dimensions. The best result already known, namely the (N - 1)-negligibility, is improved here
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