Traveling wave solutions for fractional partial differential equations arising in mathematical physics by an improved fractional Jacobi elliptic equation method

In this paper, combining with a new generalized ansätz and the fractional Jacobi elliptic equation, an improved fractional Jacobi elliptic equation method is proposed for seeking exact solutions of space‐time fractional partial differential equations. The fractional derivative used here is the modified Riemann‐Liouville derivative. For illustrating the validity of this method, we apply it to solve the space‐time fractional Fokas equation and the the space‐time fractional BBM equation. As a result, some new general exact solutions expressed in various forms including the solitary wave solutions, the periodic wave solutions, and Jacobi elliptic functions solutions for the two equations are found with the aid of mathematical software Maple. Copyright © 2016 John Wiley & Sons, Ltd.     

一个带Hardy项的椭圆方程的多重解

通过对偶变分方法证明了一个带Hardy项和临界非线性的非线性椭圆方程的非平凡解的存在性和多重性.     

带Robin边值条件的半线性奇异椭圆方程正解的存在性

本文研究了一类带Robin边值条件的半线性奇异椭圆方程．通过Hardy不等式，山路引理以及选取适当的试验函数验证局部PS条件，得到了此类方程正解的存在性这一结果．     

Multiple positive solutions for elliptic equations involving a concave term and critical Sobolev–Hardy exponent

In this paper, we establish the existence of multiple positive solutions for elliptic equations involving a concave term and critical Sobolev–Hardy exponent.     

Gelfand type elliptic problems under Steklov boundary conditions

For a Gelfand type semilinear elliptic equation we extend some known results for the Dirichlet problem to the Steklov problem. This extension requires some new tools, such as non-optimal Hardy inequalities, and discovers some new phenomena, in particular a different behavior of the branch of solutions and three kinds of blow-up for large solutions in critical growth equations. We also show that small values of the boundary parameter play against strong growth of the nonlinear source.     

Dirichlet problem for semilinear edge-degenerate elliptic equations with singular potential term

In this paper, we introduce weighted p-Sobolev spaces on manifolds with edge singularities. We give the proof for the corresponding edge type Sobolev inequality, Poincaré inequality and Hardy inequality. As an application of these inequalities, we prove the existence of nontrivial weak solutions for the Dirichlet problem of semilinear elliptic equations with singular potentials on manifolds with edge singularities.     

Multiple positive solutions for a semilinear elliptic equation with critical Sobolev exponent

In this paper, we study a class of semilinear elliptic equations with Hardy potential and critical Sobolev exponent. By means of the Ekeland variational principle and Mountain Pass theorem, multiple positive solutions are obtained.     

Symmetry and nonexistence of positive solutions of elliptic equations and systems with Hardy terms

We prove some symmetry property for equations with Hardy terms in cones, without any assumption at infinity. We also show symmetry property and nonexistence of entire solutions of some elliptic systems with Hardy weights.     

Existence and regularity of solutions to semi-linear Dirichlet problem of infinitely degenerate elliptic operators with singular potential term

In this paper, we study the Dirichlet problem for a class of semi-linear infinitely degenerate elliptic equations with singular potential term. By using the logarithmic Sobolev inequality and Hardy’s inequality, the existence and regularity of multiple nontrivial solutions have been proved.     

Positive solutions for singular elliptic equations with mixed Dirichlet‐Neumann boundary conditions

In this paper, we investigate the existence of positive solutions for singular elliptic equations with mixed Dirichlet‐Neumann boundary conditions involving Sobolev‐Hardy critical exponents and Hardy terms by using the concentration compactness principle, the strong maximum principle and the Mountain Pass lemma. We also prove, under complementary conditions, that there is no nontrivial solution if the domain is star‐shaped with respect to the origin.     

BLOW-UP SOLUTIONS FOR A CASE OF b-FAMILY EQUATIONS

In this article, we study the blow-up solutions for a case of b-family equations.Using the qualitative theory of differential equations and the bifurcation method of dynamical systems, we obtain five types of blow-up solutions: the hyperbolic blow-up solution, the fractional blow-up solution, the trigonometric blow-up solution, the first elliptic blow-up solution, and the second elliptic blow-up solution. Not only are the expressions of these blow-up solutions given, but also their relationships are discovered. In particular, it is found that two bounded solitary solutions are bifurcated from an elliptic blow-up solution.     

Existence and multiplicity of positive solutions for a class of semilinear elliptic equations involving Hardy term and Hardy-Sobolev critical exponents

Existence and multiplicity of positive solutions are studied for a class of semilinear elliptic equations with Hardy term, Hardy-Sobolev critical exponents and sublinear nonlinearity by variational methods and some analysis techniques.     

Sobolev type embedding and weak solutions with a prescribed singular set

New Sobolev type embeddings for some weighted Banach spaces are established. Using such embeddings and the singular positive radial entire solutions, we construct singular positive weak solutions with a prescribed singular set for a weighted elliptic equation. Our main results in this paper also provide positive weak solutions with a prescribed singular set to an equation with Hardy potential.     

On a semilinear elliptic equation with singular term and Hardy–Sobolev critical growth

In a previous work [6], we got an exact local behavior to the positive solutions of an elliptic equation. With the help of this exact local behavior, we obtain in this paper the existence of solutions of an equation with Hardy–Sobolev critical growth and singular term by using variational methods. The result obtained here, even in a particular case, relates with a partial (positive) answer to an open problem proposed in: A. Ferrero and F. Gazzola, Existence of solutions for singular critical growth semilinear elliptic equations, J. Differential Equations 177 , 494–522 (2001). (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Hardy-Sobolev critical singular elliptic equations with mixed Dirichlet-Neumann boundary conditions

Three important inequalities (the Poincaré, Hardy and generalized Poincaré inequalities) on the mixed boundary conditions are firstly established by some analytical techniques. Then the existence and multiplicity of positive solutions are studied for a class of semilinear elliptic equations with mixed Dirichlet-Neumann boundary conditions involving Hardy terms and Hardy-Sobolev critical exponents by using the variational methods.     

一类非线性椭圆方程正解的存在性

研究了一类非线性椭圆方程. 应用改进型Hardy不等式和变分方法, 证明了在一定条件下方程正解的存在性.     

Existence and multiplicity of solutions for semilinear elliptic equations with Hardy terms and Hardy–Sobolev critical exponents

Some existence and multiplicity results are obtained for solutions of semilinear elliptic equations with Hardy terms, Hardy–Sobolev critical exponents and superlinear nonlinearity by the variational methods and some analysis techniques.     

Fractional Hardy–Hénon equations on exterior domains

In this paper, we consider the fractional Hardy–Hénon equations with an isolated singularity. If the isolated singularity is located at the origin, we give a classification of solutions to this equation. If the isolated singularity is located at infinity, in the case of exterior domains, we provide decay estimates of solutions and their gradients at infinity. Our results are an extension of the classical work by Caffarelli, Gidas et al.     

Priori estimates and asymptotic properties of solutions for some fractional order elliptic equations

In this paper, we obtain estimates for solutions for a class of fractional order elliptic equations in different domains and boundary conditions, and prove some regularity results. Then, we study the qualitative properties of solutions with prescribed Q-curvature.     

Solutions for a singular critical growth problem with a weight

In this paper, we consider some semilinear elliptic equations with Hardy potential. By using linking theorem in [P. Rabinowitz, Minimax Methods in Critical Points Theory with Applications to Differential Equations, CBMS Reg. Conf. Ser. Math., vol. 65, Amer. Math. Soc., Providence, RI, 1986] and analyzing the effect of nonlinearities, we establish the existence of nontrivial solutions.