共查询到20条相似文献,搜索用时 8 毫秒
1.
Jesus Garcia Azorero Andrea Malchiodi Luigi Montoro Ireneo Peral 《Annales de l'Institut Henri Poincaré (C) Analyse Non Linéaire》2010
In this paper we carry on the study of asymptotic behavior of some solutions to a singularly perturbed problem with mixed Dirichlet and Neumann boundary conditions, started in the first paper [J. Garcia Azorero, A. Malchiodi, L. Montoro, I. Peral, Concentration of solutions for some singularly perturbed mixed problems: Existence results, Arch. Ration. Mech. Anal., in press]. Here we are mainly interested in the analysis of the location and shape of least energy solutions when the singular perturbation parameter tends to zero. We show that in many cases they coincide with the new solutions produced in [J. Garcia Azorero, A. Malchiodi, L. Montoro, I. Peral, Concentration of solutions for some singularly perturbed mixed problems: Existence results, Arch. Ration. Mech. Anal., in press]. 相似文献
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The existence and concentration behavior of nodal solutions are established for the equation −?2Δu+V(z)u=f(u) in Ω, where Ω is a domain in R2, not necessarily bounded, V is a positive Hölder continuous function and f∈C1 is an odd function having critical exponential growth. 相似文献
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We show that for ε small, there are arbitrarily many nodal solutions for the following nonlinear elliptic Neumann problem where Ω is a bounded and smooth domain in ℝ2 and f grows superlinearly. (A typical f(u) is f(u)= a1 u+p – a1 u-p, a1, a2 >0, p, q>1.) More precisely, for any positive integer K, there exists εK>0 such that for 0<ε<εK, the above problem has a nodal solution with K positive local maximum points and K negative local minimum points. This solution has at least K+1 nodal domains. The locations of the maximum and minimum points are related to the mean curvature on ∂Ω. The solutions are constructed as critical points of some finite dimensional reduced energy functional. No assumption on the symmetry, nor the geometry, nor the topology of the domain is needed. 相似文献
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We prove an existence result for radial solutions of a Neumann elliptic problem whose nonlinearity asymptotically lies between the first two eigenvalues. To this aim, we introduce an alternative nonresonance condition with respect to the second eigenvalue which, in the scalar case, generalizes the classical one, in the spirit of Fonda et al. (1991) [2]. Our approach also applies for nonlinearities which do not necessarily satisfy a subcritical growth assumption. 相似文献
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Solutions concentrating on higher dimensional subsets for singularly perturbed elliptic equations II
We construct spike layered solutions for the semilinear elliptic equation −ε2Δu+V(x)u=K(x)up−1 on a domain Ω⊂RN which may be bounded or unbounded. The solutions concentrate simultaneously on a finite number of m-dimensional spheres in Ω. These spheres accumulate as ε→0 at a prescribed sphere in Ω whose location is determined by the potential functions V,K. 相似文献
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The equation with boundary Dirichlet zero data is considered in a bounded domain . Under the assumption that concentrates, as , round a manifold and that f is a superlinear function, satisfying suitable growth assumptions, the existence of multiple distinct positive solutions
is proved.
Received: 19 December 2000 / Accepted: 8 May 2001 / Published online: 5 September 2002 相似文献
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MoJIAQI 《高校应用数学学报(英文版)》1996,11(2):153-158
Abstract. The singularly perturbed problems for elliptic systems in unbounded domains are considered. Under suitable conditions and by using the comparison theorem the existence and asymptotic behavior of solution for the boundary value problems studied, 相似文献
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Franco Obersnel 《Journal of Differential Equations》2010,249(7):1674-1725
We discuss existence and multiplicity of positive solutions of the prescribed mean curvature problem
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J. Chabrowski M. Willem 《Calculus of Variations and Partial Differential Equations》2002,15(4):421-431
We investigate the effect of the coefficient of the critical nonlinearity for the Neumann problem on the existence of least
energy solutions. As a by-product we establish a Sobolev inequality with interior norm.
Received: 26 April 2000 / Accepted: 25 February 2001 / Published online: 5 September 2002 相似文献
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We consider an elliptic perturbation problem in a circle by using the analytical solution that is given by a Fourier series with coefficients in terms of modified Bessel functions. By using saddle point methods we construct asymptotic approximations with respect to a small parameter. In particular we consider approximations that hold uniformly in the boundary layer, which is located along a certain part of the boundary of the domain. 相似文献
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We consider the boundary value problem Δu+up=0 in a bounded, smooth domain Ω in R2 with homogeneous Dirichlet boundary condition and p a large exponent. We find topological conditions on Ω which ensure the existence of a positive solution up concentrating at exactly m points as p→∞. In particular, for a nonsimply connected domain such a solution exists for any given m?1. 相似文献
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Yajing Zhang 《Nonlinear Analysis: Theory, Methods & Applications》2012,75(4):2047-2059
In this paper we prove the existence of two solutions for the inhomogeneous Neumann problem with critical Sobolev exponent. 相似文献
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Jianshe Yu 《Journal of Differential Equations》2009,247(2):672-684
Considered in this paper is the existence/nonexistence of periodic solutions with prescribed minimal periods to the classical forced pendulum equation,
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Jan Chabrowski Zhi-Qiang Wang 《NoDEA : Nonlinear Differential Equations and Applications》2007,13(5-6):683-697
We consider the solvability of the Neumann problem for equation (1.1) in exterior domains in both cases: subcritical and critical.
We establish the existence of least energy solutions. In the subcritical case the coefficient
b(x) is allowed to have a potential well whose steepness is controlled by a parameter λ > 0. We show that least energy solutions
exhibit a tendency to concentrate to a solution of a nonlinear problem with mixed boundary value conditions. 相似文献
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A new version of perturbation theory is developed which produces infinitely many sign-changing critical points for uneven functionals. The abstract result is applied to the following elliptic equations with a Hardy potential and a perturbation from symmetry:
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Asymptotic formulae for Green's functions for the operator -Δ in domains with small holes are obtained. A new feature of these formulae is their uniformity with respect to the independent variables. The cases of multi-dimensional and planar domains are considered. 相似文献
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We prove the uniform Hölder continuity of solutions for two classes of singularly perturbed parabolic systems. These systems arise in Bose-Einstein condensates and in competing models in population dynamics. The proof relies upon the blow up technique and the monotonicity formulas by Almgren and Alt, Caffarelli, and Friedman. 相似文献