Related papers: On the triharmonic Lane-Emden equation

We prove that the superlinear indefinite equation \begin{equation*} u" + a(t)u^{p} = 0, \end{equation*} where $p > 1$ and $a(t)$ is a $T$-periodic sign-changing function satisfying the (sharp) mean value condition $\int_{0}^{T} a(t)~\!dt <…

The purpose of this note is to establish, in terms of the primary coefficients in the framework of the tridiagonal theory developed by Delsarte and Genin in the environment of nonnegative definite Toeplitz matrices, necessary and sufficient…

In the joint work of the author with Da Lio and Rivi\`ere (Morse Index Stability for Sequences of Sacks-Uhlenbeck Maps into a Sphere) we studied the stability of the Morse index for Sacks-Uhlenbeck sequences into spheres as $p\searrow2$.…

To a complex polynomial function $f$ with arbitrary singularities we associate the number of Morse points in a general linear Morsification $f_{t} := f - t\ell$. We produce computable algebraic formulas in terms of invariants of $f$ for the…

In the Newtonian $n$-body problem for solutions with arbitrary energy, which start and end either at a total collision or a parabolic/hyperbolic infinity, we prove some basic results about their Morse and Maslov indices. Moreover for…

We classify stable and finite Morse index solutions to general semilinear elliptic equations posed in Euclidean space of dimension at most 10, or in some unbounded domains.

We prove existence results for the Lane-Emden type system \[ \begin{cases} \begin{aligned} (-\Delta)^{\alpha} u=\left| v \right|^q \\ (-\Delta)^{\beta} v= \left| u \right|^p \end{aligned} \text{ in } B_1 \subset \mathbb{R}^N \\…

By a combination of variational and topological techniques in the presence of invariant cones, we detect a new type of positive axially symmetric solutions of the Dirichlet problem for the elliptic equation $$ -\Delta u + u = a(x)|u|^{p-2}u…

We consider nonlinear second order elliptic problems of the type \[ -\Delta u=f(u) \text{ in } \Omega, \qquad u=0 \text{ on } \partial \Omega, \] where $\Omega$ is an open $C^{1,1}$-domain in $\mathbb{R}^N$, $N\geq 2$, under some general…

We derive monotone properties of positive harmonic functions on three dimensional manifolds with nonnegative scalar curvature, with an asymptotically flat end. Rigidity characterization of spatial Schwarzschild manifolds with two ends is…

We consider singular solutions to quasilinear elliptic equations under zero Dirichlet boundary condition. Under suitable assumptions on the nonlinearity we deduce symmetry and monotonicity properties of positive solutions via an improved…

In this paper, we prove stability or instability of solitons for the cubic-quintic nonlinear Schrodinger equation at every frequency. The monotonicity conjecture raised by Killip, Oh, Pocovnicu and Visan is resolved. We introduce and solve…

We develop the relative Morse index theory for linear self-adjoint operator equation without compactness assumption and give the relationship between the index defined in [44] and [45]. Then we generalize the method of saddle point…

We present new criteria on the existence of fixed points that combine some monotonicity assumptions with the classical fixed point index theory. As an illustrative application, we use our theoretical results to prove the existence of…

A priori estimates for semilinear higher order elliptic equations usually have to deal with the absence of a maximum principle. This note presents some regularity estimates for the polyharmonic Dirichlet problem that will make a distinction…

We construct ample smooth strictly plurisubharmonic non-quadratic solutions to the Monge-Amp\`ere equation on either cylindrical type domains or the whole complex Euclidean space $\mathbb C^2$. Among these, the entire solutions defined on…

We construct a monotonicity formula for the free boundary problem of the form $\Delta u=\mu$, where $\mu$ is a Radon measure. It implies that the blow up limits of solutions are homogenous functions of degree one. The first formula is new…

Following Smale, we study simple symmetric mechanical systems of $n$ point particles in the plane. In particular, we address the question of the linear and spectral stability properties of relative equilibria, which are special solutions of…

Let $Q(x,y)= 4 \frac{y^2-x^2+3}{ (x^2+y^2+3)^2}$ be the lump solution of the KP-I equation $$ \partial_x^2 (\partial_x^2 u-u + 3 u^2)-\partial_y^2 u=0.$$ We show that this solution is rigid in the following senses: the only decaying…

We are interested in finding a solution to the tensor complementarity problem with a strong M-tensor, which we call the M-tensor complementarity problem. We propose a lower dimensional linear equation approach to solve that problem. At each…