A priori bounds and multiplicity for fully nonlinear equations with quadratic growth in the gradient

We consider fully nonlinear uniformly elliptic equations with quadratic growth in the gradient, such as

$?F(x,u,Du,D^{2}u)=\lambda c(x)u+〈M(x)Du,Du〉+h(x)$

in a bounded domain with a Dirichlet boundary condition; here $\lambda \in \mathbb{R}$, $c,h\in L^p(\mathrm{\Omega })$, $p>n\ge 1$, $c?0$ and the matrix M satisfies $0<{\mu }_{1}I\le M\le {\mu }_{2}I$. Recently this problem was studied in the “coercive” case $\lambda c\le 0$, where uniqueness of solutions can be expected; and it was conjectured that the solution set is more complex for noncoercive equations. This conjecture was verified in 2015 by Arcoya, de Coster, Jeanjean and Tanaka for equations in divergence form, by exploiting the integral formulation of the problem. Here we show that similar phenomena occur for general, even fully nonlinear, equations in nondivergence form. We use different techniques based on the maximum principle.We develop a new method to obtain the crucial uniform a priori bounds, which permit to us to use degree theory. This method is based on basic regularity estimates such as half-Harnack inequalities, and on a Vázquez type strong maximum principle for our kind of equations.     

On meromorphic solutions of nonlinear partial differential equations of first order

We prove a uniqueness theorem in terms of value distribution for meromorphic solutions of a class of nonlinear partial differential equations of first order, which shows that such solutions f are uniquely determined by the zeros and poles of f−cj (counting multiplicities) for two distinct complex numbers c₁ and c₂.     

Local existence of multiple positive solutions to a singular cantilever beam equation

By constructing suitable cone and control functions, we prove some local existence theorems of positive solutions for a singular fourth-order two-point boundary value problem. In mechanics, the problem is called cantilever beam equation. Furthermore, we improve a famous method appeared in the studies of singular boundary value problems. The approximation theorem of completely continuous operators and the Guo-Krasnosel'skii fixed point theorem of cone expansion-compression type play important parts in this work.     

Multiplicity of solutions of a zero mass nonlinear equation on a Riemannian manifold

The relation between the number of solutions of a nonlinear equation on a Riemannian manifold and the topology of the manifold itself is studied. The technique is based on Ljusternik-Schnirelmann category and Morse theory.     

Semilinear elliptic problems near resonance with a nonprincipal eigenvalue

We consider the Dirichlet problem for the equation −Δu=λu±f(x,u)+h(x) in a bounded domain, where f has a sublinear growth and h∈L². We find suitable conditions on f and h in order to have at least two solutions for λ near to an eigenvalue of −Δ. A typical example to which our results apply is when f(x,u) behaves at infinity like a(x)|u|^q−2u, with M>a(x)>δ>0, and 1<q<2.     

The Full Study of Planar Quadratic Differential Systems Possessing a Line of Singularities at Infinity

In this article we make a full study of the class of non-degenerate real planar quadratic differential systems having all points at infinity (in the Poincaré compactification) as singularities. We prove that all such systems have invariant affine lines of total multiplicity 3, give all their configurations of invariant lines and show that all these systems are integrable via the method of Darboux having cubic polynomials as inverse integrating factors. After constructing the topologically distinct phase portraits in this class we give invariant necessary and sufficient conditions in terms of the 12 coefficients of the systems for the realization of each one of them and give representatives of the orbits under the action of the affine group and time rescaling. We construct the moduli space of this class for this action and give the corresponding bifurcation diagram. Dedicated to Professor Zhifen Zhang on the occasion of her 80th birthday     

Multiplicity of positive periodic solutions to superlinear repulsive singular equations

In this paper, we study positive periodic solutions to the repulsive singular perturbations of the Hill equations. It is proved that such a perturbation problem has at least two positive periodic solutions when the anti-maximum principle holds for the Hill operator and the perturbation is superlinear at infinity. The proof relies on a nonlinear alternative of Leray-Schauder type and on Krasnoselskii fixed point theorem on compression and expansion of cones.     

Existence, Multiplicity and Infinite Solvability of Positive Solutions for One-Dimensional p-Laplacian

The existence, multiplicity and infinite solvability of positive solutions are established for some two-point boundary value problems of one-dimensional p-Laplacian. In this paper, by multiplicity we mean the existence of m solutions, where m is an arbitrary natural number.     

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.     

Lattice path combinatorics and asymptotics of multiplicities of weights in tensor powers

We give asymptotic formulas for the multiplicities of weights and irreducible summands in high-tensor powers V_λ^⊗N of an irreducible representation V_λ of a compact connected Lie group G. The weights are allowed to depend on N, and we obtain several regimes of pointwise asymptotics, ranging from a central limit region to a large deviations region. We use a complex steepest descent method that applies to general asymptotic counting problems for lattice paths with steps in a convex polytope.