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1.
The main purpose of this paper is to analyze the asymptotic behavior of the radial solution of Hénon equation −Δu = |x|
α
u
p−1, u > 0, x ∈ B
R
(0) ⊂ ℝ
n
(n ⩾ 3), u = 0, x ∈ ∂B
R
(0), where $
p \to p(\alpha ) = \frac{{2(n + \alpha )}}
{{n - 2}}
$
p \to p(\alpha ) = \frac{{2(n + \alpha )}}
{{n - 2}}
from left side, α > 0. 相似文献
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3.
E.T. Karimov 《Applied Mathematics Letters》2010,23(5):517-522
The present work is devoted to the studying of a boundary-value problem with Neumann’s condition for three-dimensional elliptic equation with singular coefficients. The main result is a proof of the unique solvability of the problem considered. An energy integral method and a Green’s function method were used as the main tools in the proof of the main result. The unique solution is found in an explicit form, which contains Appel’s hypergeometric functions. 相似文献
4.
This paper is concerned with well-posedness of the Cahn–Hilliard equation subject to a class of new dynamic boundary conditions. The system was recently derived in Liu–Wu (Arch. Ration. Mech. Anal. 233 (2019), 167–247) via an energetic variational approach and it naturally fulfils three physical constraints such as mass conservation, energy dissipation and force balance. The target problem examined in this paper can be viewed as a transmission problem that consists of Cahn–Hilliard type equations both in the bulk and on the boundary. In our approach, we are able to deal with a general class of potentials with double-well structure, including the physically relevant logarithmic potential and the non-smooth double-obstacle potential. Existence, uniqueness and continuous dependence of global weak solutions are established. The proof is based on a novel time-discretization scheme for the approximation of the continuous problem. Besides, a regularity result is shown with the aim of obtaining a strong solution to the system. 相似文献
5.
We study the Dirichlet problem for the Hénon equation
where Ω is the unit ball in , with N ≥ 3, the power α is positive and is a small positive parameter. We prove that for every integer k ≥ 1 the above problem has a solution which blows up at k different points of ∂Ω as goes to zero. We also show that the ground state solution (which blows up at one point) is unique.
The first author is supported by the M.I.U.R. National Project “Metodi variazionali e topologici nello studio di fenomeni
non lineari” . The second author is supported by the M.I.U.R. National Project “Metodi variazionali ed equazioni differenziali
nonlineari”. 相似文献
6.
Pierluigi Colli Gianni Gilardi Jürgen Sprekels 《Journal of Mathematical Analysis and Applications》2014
The Cahn–Hilliard and viscous Cahn–Hilliard equations with singular and possibly nonsmooth potentials and dynamic boundary condition are considered and some well-posedness and regularity results are proved. 相似文献
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Ryuji Kajikiya 《Journal of Differential Equations》2012,252(2):1987-2003
In this paper, we study the generalized Hénon equation with a radial coefficient function in the unit ball and show the existence of a positive non-radial solution. Our result is applicable to a wide class of coefficient functions. Our theorem ensures that if the ratio of the density of the coefficient function in to that in is small enough and a is sufficiently close to 1, then a least energy solution is not radially symmetric. 相似文献
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A. S. Romanov 《Siberian Mathematical Journal》2007,48(1):142-149
For the Sobolev classes W p 1 on a “zero” cusp with a Hölder singularity at the vertex, we consider the question of compactness of the embedding of the traces of Sobolev functions into the Lebesgue classes on the boundary of the cusp. 相似文献
12.
Enrico Serra 《Calculus of Variations and Partial Differential Equations》2005,23(3):301-326
We study the Dirichlet problem in a ball for the Hénon equation with critical growth and we establish, under some conditions,
the existence of a positive, non radial solution. The solution is obtained as a minimizer of the quotient functional associated
to the problem restricted to appropriate subspaces of H01 invariant for the action of a subgroup of
. Analysis of compactness properties of minimizing sequences and careful level estimates are the main ingredients of the proof.
Received: 18 October 2003, Accepted: 5 July 2004, Published online: 3 September 2004
Mathematics Subject Classification (2000):
35J20, 35B33
This research was supported by MIUR, Project "Variational Methods and Nonlinear Differential Equations". 相似文献
13.
《Journal de Mathématiques Pures et Appliquées》2003,82(6):685-710
The ground states of the one-dimensional Falicov–Kimball model are studied in the grand canonical ensemble for large values of the interaction strength U. The quantum particle chemical potential μe is chosen in the interval −U+4<μe<0, such that, then, these states are neutral states and depend only on the sum of the two chemical potentials, μ=μi+μe. Consequences of this study are, among others, the following results. If ρ=p/q (p and q relatively prime) is a rational number we prove that, for U⩾U0(q) (where U0(q) is a specific function), there is an interval on the μ-axis, of length larger than qU−2q+3, such that for any μ in this interval, the ground state has density ρ. In this interval the ground state is unique, up to translations, and the corresponding classical particle configuration is described by the characteristic sequence associated with the rational number ρ. 相似文献
14.
Wen-Rui Shan Tian-Zhong Yan Xing Lü Min Li Bo Tian 《Communications in Nonlinear Science & Numerical Simulation》2013,18(7):1568-1575
Under investigation in this paper is the Sawada–Kotera equation with a nonvanishing boundary condition, which describes the evolution of steeper waves of shorter wavelength than those described by the Korteweg–de Vries equation does. With the binary-Bell-polynomial, Hirota method and symbolic computation, the bilinear form and N-soliton solutions for this model are derived. Meanwhile, propagation characteristics and interaction behaviors of the solitons are discussed through the graphical analysis. Via Bell-polynomial approach, the Bäcklund transformation is constructed in both the binary-Bell-polynomial and bilinear forms. Based on the binary-Bell-polynomial-type Bäcklund transformation, we obtain the Lax pair and conservation laws associated. 相似文献
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For the boundary value problem
and some more general problems the existence of any a priori given number of different positive solutions is established
under suitable conditions on q and α. Nonradial solutions to the problem are constructed for some supercritical q. Bibliography:
30 titles.
__________
Translated from Problemy Matematicheskogo Analiza, No. 35, 2007 pp. 91–110. 相似文献
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T. E. Moiseev 《Differential Equations》2009,45(10):1547-1549
We study the existence of a regular (classical) solution of the Tricomi problem for the Lavrent’ev-Bitsadze equation with mixed boundary conditions. We find conditions under which the homogeneous problem has only the zero solution and give an example in which the homogeneous Tricomi problem has a nonzero solution. We also study the solvability of the inhomogeneous Tricomi problem. 相似文献