Quasilinear elliptic equations in \mathbb{R }^{N} via variational methods and Orlicz-Sobolev embeddings

In this paper we prove the existence of a nontrivial non-negative radial solution for the quasilinear elliptic problem $$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} -\nabla \cdot \left[\phi ^{\prime }(|\nabla u|^2)\nabla u \right] +|u|^{\alpha -2}u =|u|^{s-2} u,&x\in \mathbb{R }^{N},\\ u(x) \rightarrow 0, \quad \text{ as} |x|\rightarrow \infty , \end{array} \right. \end{aligned}$$ where $N\ge 2, \phi (t)$ behaves like $t^{q/2}$ for small $t$ and $t^{p/2}$ for large $t, 1< p<q<N, 1<\alpha \le p^* q^{\prime }/p^{\prime }$ and $\max \{q,\alpha \}< s<p^*,$ being $p^*=\frac{pN}{N-p}$ and $p^{\prime }$ and $q^{\prime }$ the conjugate exponents, respectively, of $p$ and $q$ . Our aim is to approach the problem variationally by using the tools of critical points theory in an Orlicz-Sobolev space. A multiplicity result is also given.     

Strongly nonlinear multivalued elliptic equations on a bounded domain

In this work we study the existence of nontrivial solution for the following class of multivalued quasilinear problems $$\begin{aligned} \displaystyle -\text{ div } ( \phi (|\nabla u|) \nabla u) - b(u)u \in \lambda \partial F(x,u)\;\text{ in }\;\Omega , \quad u=0\; \text{ on }\;\partial \Omega \end{aligned}$$ where $\Omega \subset \mathbb{R }^N$ is a bounded domain, $N\ge 2$ and $\partial F(x,u)$ is a generalized gradient of $F(x,t)$ with respect to $t$ . The main tools utilized are Variational Methods for Locally Lipschitz Functional and a Concentration Compactness Theorem for Orlicz space.     

Nielsen coincidence,fixed point and root theories of n-valued maps

Let ${(\phi, \psi)}$ be an (m, n)-valued pair of maps ${\phi, \psi : X \multimap Y}$ , where ${\phi}$ is an m-valued map and ${\psi}$ is n-valued, on connected finite polyhedra. A point ${x \in X}$ is a coincidence point of ${\phi}$ and ${\psi}$ if ${\phi(x) \cap \psi(x) \neq \emptyset}$ . We define a Nielsen coincidence number ${N(\phi : \psi)}$ which is a lower bound for the number of coincidence points of all (m, n)-valued pairs of maps homotopic to ${(\phi, \psi)}$ . We calculate ${N(\phi : \psi)}$ for all (m, n)-valued pairs of maps of the circle and show that ${N(\phi : \psi)}$ is a sharp lower bound in that setting. Specifically, if ${\phi}$ is of degree a and ${\psi}$ of degree b, then ${N(\phi : \psi) = \frac{|an - bm|}{\langle m, n \rangle}}$ , where ${\langle m, n \rangle}$ is the greatest common divisor of m and n. In order to carry out the calculation, we obtain results, of independent interest, for n-valued maps of compact connected Lie groups that relate the Nielsen fixed point number of Helga Schirmer to the Nielsen root number of Michael Brown.     

On functions whose symmetric part of gradient agree and a generalization of Reshetnyak’s compactness theorem

We consider the following question: Given a connected open domain ${\Omega \subset \mathbb{R}^n}$ , suppose ${u, v : \Omega \rightarrow \mathbb{R}^n}$ with det ${(\nabla u) > 0}$ , det ${(\nabla v) > 0}$ a.e. are such that ${\nabla u^T(x)\nabla u(x) = \nabla v(x)^T \nabla v(x)}$ a.e. , does this imply a global relation of the form ${\nabla v(x) = R\nabla u(x)}$ a.e. in Ω where ${R \in SO(n)}$ ? If u, v are C ¹ it is an exercise to see this true, if ${u, v\in W^{1,1}}$ we show this is false. In Theorem 1 we prove this question has a positive answer if ${v \in W^{1,1}}$ and ${u \in W^{1,n}}$ is a mapping of L ^p integrable dilatation for p > n ? 1. These conditions are sharp in two dimensions and this result represents a generalization of the corollary to Liouville’s theorem that states that the differential inclusion ${\nabla u \in SO(n)}$ can only be satisfied by an affine mapping. Liouville’s corollary for rotations has been generalized by Reshetnyak who proved convergence of gradients to a fixed rotation for any weakly converging sequence ${v_k \in W^{1,1}}$ for which $$\int \limits_{\Omega} {\rm dist}(\nabla v_k, SO(n))dz \rightarrow 0 \, {\rm as} \, k \rightarrow \infty.$$ Let S(·) denote the (multiplicative) symmetric part of a matrix. In Theorem 3 we prove an analogous result to Theorem 1 for any pair of weakly converging sequences ${v_k \in W^{1,p}}$ and ${u_k \in W^{1,\frac{p(n-1)}{p-1}}}$ (where ${p \in [1, n]}$ and the sequence (u _k ) has its dilatation pointwise bounded above by an L ^r integrable function, r > n ? 1) that satisfy ${\int_{\Omega} |S(\nabla u_k) - S(\nabla v_k)|^p dz \rightarrow 0}$ as k → ∞ and for which the sign of the det ${(\nabla v_k)}$ tends to 1 in L ¹. This result contains Reshetnyak’s theorem as the special case (u _k ) ≡ Id, p = 1.     

Nonlinear superharmonic functions and supersolutions

Due to the lack of representation formulas for superharmonic functions associated with p-harmonic equations ${-\nabla \cdot(|\nabla u|^{p-2}\nabla u) = \mu}$ and their generalizations ${-\nabla \cdot A(x,\nabla u) = \mu}$ ,where ${A(x,\nabla u) \cdot \nabla u \approx | \nabla u |^{p}}$ , the interplay between nonlinear superharmonic functions and supersolutions is more important than in the linear case. Using the recent result of Kilpeläinen et. al., we establish sufficient and necessary conditions in terms of the Riesz measure μ that a p-superharmonic function is an ordinary weak supersolution. As an example we consider p-superharmonic solutions of the Poisson-type equation ${-\nabla \cdot A(x,\nabla u) = f(x)}$ .     

Nonscattering solutions and blowup at infinity for the critical wave equation

We consider the critical focusing wave equation $(-\partial _t^2+\Delta )u+u^5=0$ in ${\mathbb{R }}^{1+3}$ and prove the existence of energy class solutions which are of the form $$\begin{aligned} u(t,x)=t^\frac{\mu }{2}W(t^\mu x)+\eta (t,x) \end{aligned}$$ in the forward lightcone $\{(t,x)\in {\mathbb{R }}\times {\mathbb{R }}^3: |x|\le t, t\gg 1\}$ where $W(x)=(1+\frac{1}{3} |x|^2)^{-\frac{1}{2}}$ is the ground state soliton, $\mu $ is an arbitrary prescribed real number (positive or negative) with $|\mu |\ll 1$ , and the error $\eta $ satisfies $$\begin{aligned} \Vert \partial _t \eta (t,\cdot )\Vert _{L^2(B_t)} +\Vert \nabla \eta (t,\cdot )\Vert _{L^2(B_t)}\ll 1,\quad B_t:=\{x\in {\mathbb{R }}^3: |x|<t\} \end{aligned}$$ for all $t\gg 1$ . Furthermore, the kinetic energy of $u$ outside the cone is small. Consequently, depending on the sign of $\mu $ , we obtain two new types of solutions which either concentrate as $t\rightarrow \infty $ (with a continuum of rates) or stay bounded but do not scatter. In particular, these solutions contradict a strong version of the soliton resolution conjecture.     

Solutions of the Allen-Cahn equation which are invariant under screw-motion

Let ${\phi(x)}$ be a rational function of degree >?1 defined over a number field K and let ${\Phi_{n}(x,t) = \phi^{(n)}(x)-t \in K(x,t)}$ where ${\phi^{(n)}(x)}$ is the nth iterate of ${\phi(x)}$ . We give a formula for the discriminant of the numerator of Φ _n (x, t) and show that, if ${\phi(x)}$ is postcritically finite, for each specialization t ₀ of t to K, there exists a finite set ${S_{t_0}}$ of primes of K such that for all n, the primes dividing the discriminant are contained in ${S_{t_0}}$ .     

Inverse problem of finding n coefficients of lower derivatives in a parabolic equation

We consider the problem of reconstructing the vector function $\vec b(x) = (b_1 ,...,b_n )$ in the term $(\vec b,\nabla u)$ in a linear parabolic equation. This coefficient inverse problem is considered in a bounded domain Ω ? R ⁿ . To find the above-mentioned function $\vec b(x)$ , in addition to initial and boundary conditions we pose an integral observation of the form $\int_0^T {u(x,t)\vec \omega (t)dt = \vec \chi (x)} $ , where $\vec \omega (t) = (\omega _1 (t),...,\omega _n (t))$ is a given weight vector function. We derive sufficient existence and uniqueness conditions for the generalized solution of the inverse problem. We present an example of input data for which the assumptions of the theorems proved in the paper are necessarily satisfied.     

Equilibrium measures on saddle sets of holomorphic maps on \mathbb{P }^2

We consider the case of hyperbolic basic sets $\Lambda $ of saddle type for holomorphic maps $f:{\mathbb{P }}^2{\mathbb{C }}\rightarrow {\mathbb{P }}^2{\mathbb{C }}$ . We study equilibrium measures $\mu _\phi $ associated to a class of Hölder potentials $\phi $ on $\Lambda $ , and find the measures $\mu _\phi $ of iterates of arbitrary Bowen balls. Estimates for the pointwise dimension $\delta _{\mu _\phi }$ of $\mu _\phi $ that involve Lyapunov exponents and a correction term are found, and also a formula for the Hausdorff dimension of $\mu _\phi $ in the case when the preimage counting function is constant on $\Lambda $ . For terminal/minimal saddle sets we prove that an invariant measure $\nu $ obtained as a wedge product of two positive closed currents, is in fact the measure of maximal entropy for the restriction $f|_\Lambda $ . This allows then to obtain formulas for the measure $\nu $ of arbitrary balls, and to give a formula for the pointwise dimension and the Hausdorff dimension of $\nu $ .     

Quenched to annealed transition in the parabolic Anderson problem

We study limit behavior for sums of the form $\frac{1}{|\Lambda_{L|}}\sum_{x\in \Lambda_{L}}u(t,x),$ where the field $\Lambda_L=\left\{x\in {\bf{Z^d}}:|x|\le L\right\}$ is composed of solutions of the parabolic Anderson equation $$u(t,x) = 1 + \kappa \mathop{\int}_{0}^{t} \Delta u(s,x){\rm d}s + \mathop{\int}_{0}^{t}u(s,x)\partial B_{x}(s). $$ The index set is a box in Z ^d , namely $\Lambda_{L} = \left\{x\in {\bf Z}^{\bf d} : |x| \leq L\right\}$ and L = L(t) is a nondecreasing function $L : [0,\infty)\rightarrow {\bf R}^{+}. $ We identify two critical parameters $\eta(1) < \eta(2)$ such that for $\gamma > \eta(1)$ and L(t) = e^{γ t} , the sums $\frac{1}{|\Lambda_L|}\sum_{x\in \Lambda_L}u(t,x)$ satisfy a law of large numbers, or put another way, they exhibit annealed behavior. For $\gamma > \eta(2)$ and L(t) = e^{γ t} , one has $\sum_{x\in \Lambda_L}u(t,x)$ when properly normalized and centered satisfies a central limit theorem. For subexponential scales, that is when $\lim_{t \rightarrow \infty} \frac{1}{t}\ln L(t) = 0,$ quenched asymptotics occur. That means $\lim_{t\rightarrow \infty}\frac{1}{t}\ln\left (\frac{1}{|\Lambda_L|}\sum_{x\in \Lambda_L}u(t,x)\right) = \gamma(\kappa),$ where $\gamma(\kappa)$ is the almost sure Lyapunov exponent, i.e. $\lim_{t\rightarrow \infty}\frac{1}{t}\ln u(t,x)= \gamma(\kappa).$ We also examine the behavior of $\frac{1}{|\Lambda_L|}\sum_{x\in \Lambda_L}u(t,x)$ for L = e ^{γ t} with γ in the transition range $(0,\eta(1))$      

Existence of positive periodic solutions for second-order functional differential equations

In this paper, we show the existence of positive $T$ -periodic solutions of second-order functional differential equations $u^{\prime \prime }(t)-\rho ^2u(t)+\lambda g(t)f(u(t-\tau (t)))=0,\ \ t\in \mathbb R , $ where $\rho >0$ is a constant, $g\in C(\mathbb R ,[0,\infty ))$ , $\tau \in C(\mathbb R ,\mathbb R )$ are $T$ -periodic functions, $f\in C([0,\infty ),[0,\infty ))$ and $\lambda $ is a positive parameter. Our approach based on global bifurcation theorem.     

A maximum modulus theorem for the Oseen problem

Classical solutions of the Oseen problem are studied on an exterior domain Ω with Ljapunov boundary in R ³. It is proved a maximum modulus estimate of the following form: If ${{\bf u}\in C^2(\Omega)^3\cap C^0(\overline \Omega)^3}$ and ${p \in C^1(\Omega ), -\Delta {\bf u}+2\lambda \partial_1 {\bf u}+\nabla p=0, \nabla \cdot {\bf u}=0}$ in Ω, and if ${|{\bf u}| \le M}$ on ${\partial \Omega , \limsup |{\bf u}({\bf x})|\le M}$ as ${|{\bf x}|\to \infty }$ , then ${|{\bf u}({\bf x})|\le c M}$ in Ω. Here the constant c depends only on Ω and λ.     

Reducibility and Unitarily Equivalence for a Class of Analytic Multipliers on the Dirichlet Space

In this paper, we first prove that if $\phi $ is a finite Blaschke product with $N=2,3$ zeros, then $M_\phi $ is reducible on the Dirichlet space if and only if $\phi $ is equivalent to $z^N$ . Also, we prove that $M_\phi $ is unitary equivalent to Dirichlet shift of multiplicity $N$ if and only if $\phi =\lambda z^N$ for some unimodular constant $\lambda $ .     

Stochastics and thermodynamics for equilibrium measures of holomorphic endomorphisms on complex projective spaces

It was proved by Urbański and Zdunik (Fund Math 220:23–69, 2013) that for every holomorphic endomorphism $f:{{\mathbb { P}}}^k\rightarrow {{\mathbb { P}}}^k$ of a complex projective space ${{\mathbb { P}}}^k,k\ge 1$ , there exists a positive number $\kappa _f>0$ such that if $J$ is the Julia set of $f$ (i.e. the support of the maximal entropy measure) and $\phi :J\rightarrow {\mathbb R}$ is a Hölder continuous function with $\sup (\phi )-\inf (\phi )<\kappa _f$ (pressure gap), then $\phi $ admits a unique equilibrium state $\mu _\phi $ on $J$ . In this paper we prove that the dynamical system ( $f,\mu _\phi $ ) enjoys exponential decay of correlations of Hölder continuous observables as well as the Central Limit Theorem and the Law of Iterated Logarithm for the class of these variables that, in addition, satisfy a natural co-boundary condition. We also show that the topological pressure function $t\mapsto P(t\phi )$ is real-analytic throughout the open set of all parameters $t$ for which the potentials $t\phi $ have pressure gaps.     

Local behaviour of singular solutions for nonlinear elliptic equations in divergence form

We consider the following class of nonlinear elliptic equations $$\begin{array}{ll}{-}{\rm div}(\mathcal{A}(|x|)\nabla u) +u^q=0\quad {\rm in}\; B_1(0)\setminus\{0\}, \end{array}$$ where q > 1 and ${\mathcal{A}}$ is a positive C ¹(0,1] function which is regularly varying at zero with index ${\vartheta}$ in (2?N,2). We prove that all isolated singularities at zero for the positive solutions are removable if and only if ${\Phi\not\in L^q(B_1(0))}$ , where ${\Phi}$ denotes the fundamental solution of ${-{\rm div}(\mathcal{A}(|x|)\nabla u)=\delta_0}$ in ${\mathcal D'(B_1(0))}$ and δ₀ is the Dirac mass at 0. Moreover, we give a complete classification of the behaviour near zero of all positive solutions in the more delicate case that ${\Phi\in L^q(B_1(0))}$ . We also establish the existence of positive solutions in all the categories of such a classification. Our results apply in particular to the model case ${\mathcal{A}(|x|)=|x|^\vartheta}$ with ${\vartheta\in (2-N,2)}$ .     

Beyond the Trudinger-Moser supremum

Let Ω be a bounded, smooth domain in ${\mathbb{R}^2}$ . We consider the functional $$I(u) = \int_\Omega e^{u^2}\,dx$$ in the supercritical Trudinger-Moser regime, i.e. for ${\int_\Omega |\nabla u|^2dx > 4\pi}$ . More precisely, we are looking for critical points of I(u) in the class of functions ${u \in H_0^1 (\Omega )}$ such that ${\int_\Omega |\nabla u|^2 \, dx = 4\, \pi \, k\, (1+\alpha)}$ , for small α > 0. In particular, we prove the existence of 1-peak critical points of I(u) with ${\int_\Omega |\nabla u|^2dx = 4\pi(1 + \alpha)}$ for any bounded domain Ω, 2-peak critical points with ${\int_\Omega |\nabla u|^2dx = 8\pi(1 + \alpha)}$ for non-simply connected domains Ω, and k-peak critical points with ${\int_\Omega |\nabla u|^2 dx = 4k \pi(1 + \alpha)}$ if Ω is an annulus.     

Functional inequalities related to Mahler’s conjecture

Extending a result of Meyer and Reisner (Monatsh Math 125:219–227, 1998), we prove that if ${g: \mathbb{R}\to \mathbb{R}_+}$ is a function which is concave on its support, then for every m > 0 and every ${z\in\mathbb{R}}$ such that g(z) > 0, one has $$ \int\limits_{\mathbb{R}} g(x)^mdx\int\limits_{\mathbb{R}} (g^{*z}(y))^m dy\ge \frac{(m+2)^{m+2}}{(m+1)^{m+3}},$$ where for ${y\in \mathbb{R}}$ , ${g^{*z}(y)=\inf_x \frac{(1-(x-z)y)_+}{g(x)}}$ . It is shown how this inequality is related to a special case of Mahler’s conjecture (or inverse Santaló inequality) for convex bodies. The same ideas are applied to give a new (and simple) proof of the exact estimate of the functional inverse Santaló inequality in dimension 1 given in Fradelizi and Meyer (Adv Math 218:1430–1452, 2008). Namely, if ${\phi:\mathbb{R}\to\mathbb{R}\cup\{+\infty\}}$ is a convex function such that ${0 < \int e^{-\phi} < +\infty}$ then, for every ${z\in\mathbb{R}}$ such that ${\phi(z) < +\infty}$ , one has $$ \int\limits_{\mathbb{R}}e^{-\phi}\int\limits_{\mathbb{R}} e^{-\mathcal{L}^z\phi}\ge e,$$ where ${\mathcal {L}^z\phi}$ is the Legendre transform of ${\phi}$ with respect to z.     

Radial solutions of Neumann problems involving mean extrinsic curvature and periodic nonlinearities

We show that if ${{\mathcal A} \subset \mathbb{R}^N}$ is an annulus or a ball centered at zero, the homogeneous Neumann problem on ${{\mathcal A}}$ for the equation with continuous data $$\nabla \cdot \left(\frac{\nabla v}{\sqrt{1 - |\nabla v|^2}} \right) = g(|x|,v) + h(|x|)$$ has at least one radial solution when g(|x|,·) has a periodic indefinite integral and ${\int_{\mathcal A} h(|x|)\,{\rm{d}}x = 0.}$ The proof is based upon the direct method of the calculus of variations, variational inequalities and degree theory.     

Comparison of two regulatory results on functional equations with few parameters

Two methods to prove regularity properties of the linear functional equation $$f(x)=h_0(x,y)+\sum_{j=1}^n h_j(x,y)f(x+g_j(y)), $$ where ${(x,y) \in D \subset \mathbb{R}^r \times \mathbb{R}^s}$ , ${x \in \mathbb{R}^r}$ and ${y \in \mathbb{R}^s}$ , with few parameters i.e. allowing 1 ?? s < r are examined. It is proved that??under certain conditions, for some class of equations and in some sense??they are equivalent.     

Biharmonic maps from a complete Riemannian manifold into a non-positively curved manifold

We consider biharmonic maps $\phi :(M,g)\rightarrow (N,h)$ from a complete Riemannian manifold into a Riemannian manifold with non-positive sectional curvature. Assume that $ p $ satisfies $ 2\le p <\infty $ . If for such a $ p $ , $\int _M|\tau (\phi )|^{ p }\,\mathrm{d}v_g<\infty $ and $\int _M|\,\mathrm{d}\phi |^2\,\mathrm{d}v_g<\infty ,$ where $\tau (\phi )$ is the tension field of $\phi $ , then we show that $\phi $ is harmonic. For a biharmonic submanifold, we obtain that the above assumption $\int _M|\,\mathrm{d}\phi |^2\,\mathrm{d}v_g<\infty $ is not necessary. These results give affirmative partial answers to the global version of generalized Chen’s conjecture.