Existence and asymptotic behavior of radially symmetric ground states of quasi-linear singular elliptic equations

Existence and asymptotic behavior of entire positive solutions of a class of quasi-linear elliptic equation is obtained. Under several hypotheses on the ρ(x) and f(r), we obtain the existence of positive entire solution. Asymptotic behavior is discussed by constructing an upper solution. The results of this paper is new and extend previously known results.     

Multiple existence of non-radial solutions with group invariance for sublinear elliptic equations

We deal with sublinear elliptic equations in a ball and prove the existence of infinitely many solutions which are not radially symmetric but G invariant. Here G is any closed subgroup of the orthogonal group and is not transitive on the unit sphere.     

Elliptic systems involving multiple critical nonlinearities and symmetric multi-polar potentials

In this paper,a system of elliptic equations is investigated,which involves multiple critical Sobolev exponents and symmetric multi-polar potentials.By variational methods and analytic techniques,the relevant best constants are studied and the existence of(Zk×SO(N.2))2-invariant solutions to the system is established.     

A strengthening of the interior Hölder continuity property for solutions of the Dirichlet problem for a second-order elliptic equation

The classical solution of the Dirichlet problem with a continuous boundary function for a linear elliptic equation with Hölder continuous coefficients and right-hand side satisfies the interior Schauder estimates describing the possible increase of the solution smoothness characteristics as the boundary is approached, namely, of the solution derivatives and their difference ratios in the corresponding Hölder norm. We prove similar assertions for the generalized solution with some other smoothness characteristics. In contrast to the interior Schauder estimates for classical solutions, our established estimates for the differential characteristics imply the continuity of the generalized solution in a sense natural for the problem (in the sense of (n-1)-dimensional continuity) up to the boundary of the domain in question. We state the global properties in terms of the boundedness of the integrals of the square of the difference between the solution values at different points with respect to especially normalized measures in a certain class.     

Multiple symmetric results for singular quasilinear elliptic systems with critical homogeneous nonlinearity

This paper deals with the existence and multiplicity of symmetric solutions for a class of singular quasilinear elliptic systems with critical homogeneous nonlinearity in a bounded symmetric domain. Applying variational methods and the symmetric criticality principle of Palais, we establish several existence and multiplicity results of G‐symmetric solutions under some appropriate assumptions on the weighted functions and the parameters. Copyright © 2016 John Wiley & Sons, Ltd.     

Maximal- and minimal symmetric solutions of fully fuzzy linear systems

In this paper, we shall propose a new method to obtain symmetric solutions of a fully fuzzy linear system (FFLS) based on a 1-cut expansion. To this end, we solve the 1-cut of a FFLS (in the present paper, we assumed that the 1-cut of a FFLS is a crisp linear system or equivalently, the matrix coefficient and right hand side have triangular shapes), then some unknown symmetric spreads are allocated to each row of a 1-cut of a FFLS. So, after some manipulations, the original FFLS is transformed to solving 2n linear equations to find the symmetric spreads. However, our method always give us a fuzzy number vector solution. Moreover, using the proposed method leads to determining the maximal- and minimal symmetric solutions of the FFLS which are placed in a Tolerable Solution Set and a Controllable Solution Set, respectively. However, the obtained solutions could be interpreted as bounded symmetric solutions of the FFLS which are useful for a large number of multiplications existing between two fuzzy numbers. Finally, some numerical examples are given to illustrate the ability of the proposed method.     

On the symmetric solutions of a linear matrix equation

A formula for the partitioned minimum-norm reflexive generalized inverse is applied to find the general symmetric solution X to the matrix equation AX=B. Also the dimension of the space of symmetric solutions is established.     

Nodal and positive solutions for singular semilinear elliptic equations with critical exponents in symmetric domains

In this paper, we consider the semilinear elliptic problem in Ω, u=0 on ∂Ω, where Ω is a smooth bounded domain in RN, N?4, , is the critical Sobolev exponent, K(x) is a continuous function. When Ω and K(x) are invariant under a group of orthogonal transformations, we prove the existence of nodal and positive solutions for 0<λ<λ₁, where λ₁ is the first Dirichlet eigenvalue of on Ω.     

Sobolev spaces of symmetric functions and applications

We prove sharp pointwise estimates for functions in the Sobolev spaces of radial functions defined in a ball. As a consequence, we obtain some imbeddings of such Sobolev spaces in weighted Lq-spaces. We also prove similar imbeddings for Sobolev spaces of functions with partial symmetry. Our techniques lead to new Hardy type inequalities. It is important to observe that we do not require any vanishing condition on the boundary to obtain all our estimates. We apply these imbeddings to obtain radial solutions and partially symmetric solutions for a biharmonic equation of the Hénon type under both Dirichlet and Navier boundary conditions. The delicate question of the regularity of these solutions is also established.     

Barenblatt solution and spherically symmetric solutions of the porous media equation

We consider the Cauchy problem of the porous media equation. We show that it is spherically symmetric solution has the same property as Barenblatt solution, with respct to some regularity property.     

Uniqueness of a positive radially symmetric Solution of the Dirichlet problem for certain nonlinear differential equation of the second order in a ball

We consider the Dirichlet problem

$u_\Gamma = 0$

for the nonlinear differential equation

$\Delta u + \left| x \right|^m \left| u \right|^p = 0, x \in S,$

with constant m ≥ 0 and p > 1 in the unit ball S = {x ∈ R ⁿ : |x| < 1}(n ≥ 3) with the boundary Γ. We prove that with p ≤ m+n/n?2 this problem has a unique positive radially symmetric solution.

     

Infinitely many solutions for symmetric and non-symmetric elliptic systems

We study an elliptic system equivalent to a fourth order elliptic equation. By using variational and perturbative methods, we prove the existence of infinitely many solutions both in the symmetric and in the non-symmetric case.     

Symmetry of translating solutions to mean curvature flows

First, we review the authors' recent results on translating solutions to mean curvature flows in Euclidean space as well as in Minkowski space, emphasizing on the asymptotic expansion of rotationally symmetric solutions. Then we study the sufficient condition for which the translating solution is rotationally symmetric. We will use a moving plane method to show that this condition is optimal for the symmetry of solutions to fully nonlinear elliptic equations without ground state condition.     

Rotationally symmetric translating soliton of H~k-flow

This paper mainly considers the translating soliton of H k-flow for k > 0.We give the asymptotic expression of the entire rotationally symmetric translating soliton,and obtain non-convex Wing-like solution as well as two barrier solutions.Moreover,we show that the solution with polynomial growth keeps its growth rate when evolution.     

Behavior at infinity of solutions of an equation of Osserman-Keller type

We study the behavior at infinity of solutions of equations of the form Δu=u^p, where p>1, in dimensions n?3. In particular we extend results proved by Loewner and Nirenberg in Contribution to Analysis, 1974, pp. 245-272 for the case p=(n+2)/(n−2), n?3, to values of p in the range p>n/(n−2), n?3.     

Rigorous results in selection of steady needle crystals

This paper concerns the existence of solutions to a steady needle crystal growth problem in a one-sided model. We rigorously prove that for small nonzero anisotropy γ, analytic symmetric needle crystal solutions exist in the limit of surface tension ε² if only if the stokes constant S for a relatively simple nonlinear differential equation is zero. This Stokes constant S depends on the parameter β=2^9/7γε^−8/7 and earlier numerical calculations by a number of investigators have shown this to be zero for a discrete set of values of β. It is also proven that for γ=0, there can be no symmetric needle crystal solution in the considered space.The methodology consists of two steps. First, the original problem is reduced to a weak half-strip problem for any γ in a compact set of [0,1) by relaxation of the symmetry condition. The weak problem is shown to have a unique solution in the function space considered for any γ∈[0,γ_m] for some γ_m>0. When a symmetry is invoked, the weak problem is shown equivalent to the original needle crystal problem. Next, by considering the behavior of the solution in neighborhood of an appropriate complex turning point for γ∈(0,γ_m], we extract an exponentially small term in ε as ε→0⁺ that generally violates the symmetric condition. We prove that the symmetry condition is satisfied for small ε when the parameter β is constrained appropriately.     

Existence of sign-changing solutions for semilinear singular elliptic equations with critical Sobolev exponents

We consider the semilinear elliptic problem in Ω, u=0 on ∂Ω, where 0∈Ω is a smooth bounded domain in RN, N?4, , is the critical Sobolev exponent, f(x,⋅) has subcritical growth at infinity, K(x)>0 is continuous. We prove the existence of sign-changing solutions under different assumptions when Ω is a usual domain and a symmetric domain, respectively.     

A self-similar solution to the equation of gas filtration in a spherically symmetric porous medium

We consider the Cauchy problem for the Boussinesq equation which describes filtration of a gas in a spherically symmetric porous medium. For the self-similar solution to this problem we construct a formal in the neighborhood of the point r → ∞ expansion and a convergent near r = 0 one.     

Least-squares solutions to the matrix equations AX = B and XC = D

The least-squares solution and the least-squares symmetric solution with the minimum-norm of the matrix equations AX = B and XC = D are considered in this paper. By the matrix differentiation and the spectral decomposition of matrices, an explicit representation of such solution is given.     

Threshold phenomena for symmetric decreasing solutions of reaction-diffusion equations

We study the long time behavior of solutions of the Cauchy problem for nonlinear reaction-diffusion equations in one space dimension with the nonlinearity of bistable, ignition or monostable type. We prove a one-to-one relation between the long time behavior of the solution and the limit value of its energy for symmetric decreasing initial data in L ² under minimal assumptions on the nonlinearities. The obtained relation allows to establish sharp threshold results between propagation and extinction for monotone families of initial data in the considered general setting.