Bifurcation and one-sign solutions of the pp-Laplacian involving a nonlinearity with zeros

In this paper, we use bifurcation method to investigate the existence and multiplicity of one-sign solutions of the $p$-Laplacian involving a linear/superlinear nonlinearity with zeros. To do this, we first establish a bifurcation theorem from infinity for nonlinear operator equation with homogeneous operator. To deal with the superlinear case, we establish several topological results involving superior limit

Br\"uck Conjecture with hyper-order less than one

In this paper we affirm Br\"{u}ck conjecture provided $f$ is of hyper-order less than one by studying the infinite hyper-order of solutions of a complex differential equation

Periodic Solutions of the Planar N-Center Problem with topological constraints

In the planar $N$-center problem, for a non-trivial free homotopy class of the configuration space satisfying certain mild condition, we show that there is at least one collision free $T$-periodic solution for any positive $T.$ We use the direct method of calculus of variations and the main difficulty is to show that minimizers under certain topological constraints are free of collision.Comment: 34 pages, 5 figures, major revision in section 4, minor revision in section

Application of Morse index in weak force NN-body problem

Due to collision singularities, the Lagrange action functional of the N-body problem in general is not differentiable. Because of this, the usual critical point theory can not be applied to this problem directly. Following ideas from \cite{BR91}, \cite{Tn93a} and \cite{ABT06}, we introduce a notion called weak critical point for such an action functional, as a generalization of the usual critical point. A corresponding definition of Morse index for such a weak critical point will also be given. Moreover it will be shown that the Morse index gives an upper bound of the number of possible binary collisions in a weak critical point of the $N$-body problem with weak force potentials including the Newtonian potential.Comment: 17 pages. Some minor change

Connecting planar linear chains in the spatial NN-body problem

The family of planar linear chains are found as collision-free action minimizers of the spatial $N$-body problem with equal masses under $D_N$ or D_N \times \zz_2-symmetry constraint and different types of topological constraints. This generalizes a previous result by the author in \cite{Y15c} for the planar $N$-body problem. In particular, the monotone constraints required in \cite{Y15c} are proven to be unnecessary, as it will be implied by the action minimization property. For each type of topological constraints, by considering the corresponding action minimization problem in a coordinate frame rotating around the vertical axis at a constant angular velocity \om, we find an entire family of simple choreographies (seen in the rotating frame), as \om changes from $0$ to $N$. Such a family starts from one planar linear chain and ends at another (seen in the original non-rotating frame). The action minimizer is collision-free, when \om=0 or $N$, but may contain collision for 0 < \om < N. However all possible collisions must be binary and each collision solution is $C^0$ block-regularizable. Moreover for certain types of topological constraints, based on results from \cite{BT04} and \cite{CF09}, we show that when \om belongs to some sub-intervals of $[0, N]$, the corresponding minimizer must be a rotating regular $N$-gon contained in the horizontal plane. As a result, this generalizes Marchal's $P_{12}$ family of the three body problem to arbitrary $N \ge 3$.Comment: 32 pages, 5 figures. Fixed a mistake and added one more figur

Dipole-Dipole Correlations for the sine-Gordon Model

We consider the dipole-dipole correlations for the two-dimensional Coulomb gas/sine-Gordon model for $\beta> 8\pi$ by a renormalization group method. First we re-establish the renormalization group analysis for the partition function using finite range decomposition of the covariance. Then we extend the analysis to the correlation functions. Finally, we show a power-law decay characteristic of the dipole gas

Eigenvalue, global bifurcation and positive solutions for a class of fully nonlinear problems

In this paper, we shall study global bifurcation phenomenon for the following Kirchhoff type problem \begin{equation} \left\{ \begin{array}{l} -\left(a+b\int_\Omega \vert \nabla u\vert^2\,dx\right)\Delta u=\lambda u+h(x,u,\lambda)\,\,\text{in}\,\, \Omega,\\ u=0~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\text{on}\,\,\Omega. \end{array} \right.\nonumber \end{equation} Under some natural hypotheses on $h$, we show that $\left(a\lambda_1,0\right)$ is a bifurcation point of the above problem. As applications of the above result, we shall determine the interval of $\lambda$, in which there exist positive solutions for the above problem with $h(x,u;\lambda)=\lambda f(x,u)-\lambda u$, where $f$ is asymptotically linear at zero and is asymptotically 3-linear at infinity. To study global structure of bifurcation branch, we also establish some properties of the first eigenvalue for a nonlocal eigenvalue problem. Moreover, we also provide a positive answer to an open problem involving the case of $a=0$.Comment: 20 page

Two Whyburn type topological theorems and its applications to Monge-Amp\`{e}re equations

In this paper we correct a gap of Whyburn type topological lemma and establish two superior limit theorems. As the applications of our Whyburn type topological theorems, we study the following Monge-Amp\`{e}re equation \begin{eqnarray} \left\{ \begin{array}{lll} \det\left(D^2u\right)=\lambda^N a(x)f(-u)\,\, &\text{in}\,\, \Omega,\\ u=0~~~~~~~~~~~~~~~~~~~~~~\,\,&\text{on}\,\, \partial \Omega. \end{array} \right.\nonumber \end{eqnarray} We establish global bifurcation results for the problem. We find intervals of $\lambda$ for the existence, multiplicity and nonexistence of strictly convex solutions for this problem.Comment: arXiv admin note: substantial text overlap with arXiv:1207.666

Parameter optimization in differential geometry based solvation models

Differential geometry (DG) based solvation models are a new class of variational implicit solvent approaches that are able to avoid unphysical solvent-solute boundary definitions and associated geometric singularities, and dynamically couple polar and nonpolar interactions in a self-consistent framework. Our earlier study indicates that DG based nonpolar solvation model outperforms other methods in nonpolar solvation energy predictions. However, the DG based full solvation model has not shown its superiority in solvation analysis, due to its difficulty in parametrization, which must ensure the stability of the solution of strongly coupled nonlinear Laplace-Beltrami and Poisson-Boltzmann equations. In this work, we introduce new parameter learning algorithms based on perturbation and convex optimization theories to stabilize the numerical solution and thus achieve an optimal parametrization of the DG based solvation models. An interesting feature of the present DG based solvation model is that it provides accurate solvation free energy predictions for both polar and nonploar molecules in a unified formulation. Extensive numerical experiment demonstrates that the present DG based solvation model delivers some of the most accurate predictions of the solvation free energies for a large number of molecules.Comment: 19 pages, 12 figures, convex optimizatio

Index Theory for Zero Energy Solutions of the Planar Anisotropic Kepler Problem

In the variational study of singular Lagrange systems, the zero energy solutions play an important role. In this paper we find a simple way of computing the Morse indices of these solutions for the planar anisotropic Kepler problem. In particular an interesting connection between the Morse indices and the oscillating behaviors of these solutions discovered by the physicist M. Gutzwiller is established.Comment: Revision based on referees reports. Accepted by Communications in Mathematical Physics(CMP