Sturm-Liouville边值问题解的非存在性、存在性和多重性结果

本文研究下述Sturm-Liouville边值问题利用Schauder不动点定理、上下解方法和Leray-Schauder映射度理论,获得了解的非存在性、存在性和多重性结果.其中一些是全新的结果,另一些则扩展、改进和完善了由Erbe,Wang,Hai,Lee和Lin所获得的结果.     

Entire radial solutions of elliptic systems and inequalities of the mean curvature type

In this paper we study first nonexistence of radial entire solutions of elliptic systems of the mean curvature type with a singular or degenerate diffusion depending on the solution u. In particular we extend a previous result given in [R. Filippucci, Nonexistence of radial entire solutions of elliptic systems, J. Differential Equations 188 (2003) 353-389]. Moreover, in the scalar case we obtain nonexistence of all entire solutions, radial or not, of differential inequalities involving again operators of the mean curvature type and a diffusion term. We prove that in the scalar case, nonexistence of entire solutions is due to the explosion of the derivative of every nonglobal radial solution in the right extremum of the maximal interval of existence, while in that point the solution is bounded. This behavior is qualitatively different with respect to what happens for the m-Laplacian operator, studied in [R. Filippucci, Nonexistence of radial entire solutions of elliptic systems, J. Differential Equations 188 (2003) 353-389], where nonexistence of entire solutions is due, even in the vectorial case, to the explosion in norm of the solution at a finite point. Our nonexistence theorems for inequalities extend previous results given by Naito and Usami in [Y. Naito, H. Usami, Entire solutions of the inequality div(A(|Du|)Du)?f(u), Math. Z. 225 (1997) 167-175] and Ghergu and Radulescu in [M. Ghergu, V. Radulescu, Existence and nonexistence of entire solutions to the logistic differential equation, Abstr. Appl. Anal. 17 (2003) 995-1003].     

具非线性对流项扩散方程的源型解

研究主部为热传导算子的拟线性抛物型方程Cauchy问题:u_t=u_(xx) (u~n)_x,(x,t)∈S=R×(0,∞),u(x,0)=δ(x),x∈■在一维情形下源型解的存在性,唯一性,不存在性,解的渐近性和相似源型解等问题.在研究过程中,找到了一个n的临界值,即n_0=3.当0≤n相似文献   

THE STEP-TRANSITION OPERATORS FOR MULTI-STEP METHODS OF ODE'S

1.'IntroductionThedisad~ageofsymplecticmethodsinusingtheinformationfrompasttimestepsleadstotheirneedingmorefunctionevaluationthannonsymplecticmethods.Thisdisadvantagecanbeovercomeifonecouldconstructsymplecticmulti-stepmethods.But'theaestProblemshouldbesolvedistogiveoutthedefinitionofsymplecticmultistepmethod.Ulltilnow,apopularideaisthatanm-stepmethodonMmaybewrittenasaone-stepmethodonMa.Inpaper12,71,theauthorshaveinvestigatedthecircumstanceunderWhichadifferenceschemecanpreservetheproductsympl…     

Potential well method for Cauchy problem of generalized double dispersion equations

In this paper we study Cauchy problem of generalized double dispersion equations utt−uxx−uxxtt+uxxxx=f(u)_xx, where f(u)=p|u|, p>1 or u2k, . By introducing a family of potential wells we not only get a threshold result of global existence and nonexistence of solutions, but also obtain the invariance of some sets and vacuum isolating of solutions. In addition, the global existence and finite time blow up of solutions for problem with critical initial conditions E(0)=d, I(u₀)?0 or I(u₀)<0 are proved.     

On biharmonic submanifolds in non-positively curved manifolds

In the biharmonic submanifolds theory there is a generalized Chen’s conjecture which states that biharmonic submanifolds in a Riemannian manifold with non-positive sectional curvature must be minimal. This conjecture turned out false by a counter example of Y.L. Ou and L. Tang in Ou and Tang (2012). However it remains interesting to find out sufficient conditions which guarantee this conjecture to be true. In this note we prove that:1. Any complete biharmonic submanifold (resp. hypersurface) $(M,g)$ in a Riemannian manifold $(N,h)$ with non-positive sectional curvature (resp. Ricci curvature) which satisfies an integral condition: for some $p\in (0,+\infty )$, ${\int }_M{\left|\overrightarrow{H}\right|}^{p}d{\mu }_g<+\infty$, where $\overrightarrow{H}$ is the mean curvature vector field of $M\hookrightarrow N$, must be minimal. This generalizes the recent results due to N. Nakauchi and H. Urakawa in Nakauchi and Urakawa (2013, 2011).2. Any complete biharmonic submanifold (resp. hypersurface) in a Riemannian manifold of at most polynomial volume growth whose sectional curvature (resp. Ricci curvature) is non-positive must be minimal.3. Any complete biharmonic submanifold (resp. hypersurface) in a non-positively curved manifold whose sectional curvature (resp. Ricci curvature) is smaller than $-ϵ$ for some $ϵ>0$ which satisfies that ${\int }_{B_{\rho }\left(x_0\right)}{\left|\overrightarrow{H}\right|}^{p+2}d{\mu }_g(p\ge 0)$ is of at most polynomial growth of $\rho$, must be minimal.We also consider $\epsilon$-superbiharmonic submanifolds defined recently in Wheeler (2013) by G. Wheeler and prove similar results for $\epsilon$-superbiharmonic submanifolds, which generalize the result in Wheeler (2013).     

On entire solutions of degenerate elliptic differential inequalities with nonlinear gradient terms

In this paper we give sufficient conditions for the nonexistence of nonnegative nontrivial entire weak solutions of p-Laplacian elliptic inequalities, with possibly singular weights and gradient terms, of the form , under the main request that h and are continuous on R⁺. We achieve our conclusions introducing a generalized version of the well-known Keller-Osserman condition.