Asymptotic behavior on the Hénon equation with supercritical exponent

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.     

On a boundary problem with Neumann’s condition for 3D singular elliptic equations

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.     

On a transmission problem for equation and dynamic boundary condition of Cahn–Hilliard type with nonsmooth potentials

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.     

Multi-peak solutions for the Hénon equation with slightly subcritical growth

We study the Dirichlet problem for the Hénon equation

On the Cahn–Hilliard equation with dynamic boundary conditions and a dominating boundary potential

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.     

Non-radial least energy solutions of the generalized Hénon equation

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 $|x|<a$ to that in $a<|x|<1$ is small enough and a is sufficiently close to 1, then a least energy solution is not radially symmetric.     

On the traces of Sobolev functions on the boundary of a cusp with a Hölder singularity

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.     

Non radial positive solutions for the Hénon equation with critical growth

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".     

On the ground states of the one-dimensional Falicov–Kimball model

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 μ_e is chosen in the interval −U+4<μ_e<0, such that, then, these states are neutral states and depend only on the sum of the two chemical potentials, μ=μ_i+μ_e. 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.     

Analytic study on the Sawada–Kotera equation with a nonvanishing boundary condition in fluids

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.     

Multiplicity of solutions to the Dirichlet problem for generalized Hénon equation

For the boundary value problem

On the solvability of the tricomi problem for the Lavrent’ev-Bitsadze equation with mixed boundary conditions

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.