arXiv subjects

Shibo Liu

Publications and source records attributed to Shibo Liu.

22 records · Page 2Linked to original sources

arXiv2017

On the change of variables formula for multiple integrals

In this paper, we develop an elementary proof of the change of variables in multiple integrals. Our proof is based on an induction argument. Assuming the formula for (m-1)-integrals, we define the integral over hypersurface in Rm, establish the divergent theorem and then use the divergent theorem to prove the formula for m-integrals. In addition to its simplicity, an advantage of our approach is that it yields the Brouwer Fixed Point Theorem as a corollary.

math.CA↗

arXiv2014

Existence results for fractional p-Laplacian problems via Morse theory

We investigate a class of quasi-linear nonlocal problems, including as a particular case semi-linear problems involving the fractional Laplacian and arising in the framework of continuum mechanics, phase transition phenomena, population dynamics and game theory. Under different growth assumptions on the reaction term, we obtain various existence as well as finite multiplicity results by means of variational and topological methods and, in particular, arguments from Morse theory.

math.AP↗

arXiv2013

Standing waves with large frequency for 4-superlinear Schrödinger-Poisson systems

We consider standing waves with frequency $ω$ for 4-superlinear Schrödinger-Poisson system. For large $ω$ the problem reduces to a system of elliptic equations in $\mathsf{R}^3$ with potential indefinite in sign. The variational functional does not satisfy the mountain pass geometry. The nonlinearity considered here satisfies a condition which is much weaker than the classical (AR) condition and the condition (Je) of Jeanjean. We obtain nontrivial solution and, in case of odd nonlinearity an unbounded sequence of solutions via the local linking theorem and the fountain theorem, respectively.

math.AP↗

arXiv2012

Homology of saddle point reduction and applications to resonant elliptic systems

In the setting of saddle point reduction, we prove that the critical groups of the original functional and the reduced functional are isomorphic. As application, we obtain two nontrivial solutions for elliptic gradient systems which may be resonant both at the origin and at infinity. The difficulty that the variational functional does not satisfy the Palais-Smale condition is overcame by taking advantage of saddle point reduction. Our abstract results on critical groups are crucial.

math.AP↗