Multiplicity of solutions of weighted (p,q)-Laplacian with small source

In this article, we study the existence of infinitely many solutions to the degenerate quasilinear elliptic system-div(h_1(x)|▽u|~(p-2)▽u)=d(x)|u|~(r-2)u+G_u(x,u,v) in Ω,-div(h_2(x)|▽u|~(p-2)▽v)=f(x)|v|~(s-2)v + G_u(x,u,v) in Ω,u=v=0 on ■Ω where Ω is a bonded domain in R~N with smooth boundary ■Ω,N≥2,1 r p ∞,1 s q ∞; h_1(x) and h_2(x) are allowed to have "essential" zeroes at some points inΩ; d(x)|u|~(r-2)u and f(x)|v|~(s-2)v are small sources with Gu(x,u,v), Gv(x,u,v) being their high-order perturbations with respect to(u,v) near the origin, respectively.     

Global existence and stabilization in a degenerate chemotaxis-Stokes system with mildly strong diffusion enhancement

A class of chemotaxis-Stokes systems generalizing the prototype

$\{\begin{array}{ccc}\hfill n_t+u?\mathrm{?}n& \hfill =\hfill & \mathrm{?}?\left(n^{m?1}\mathrm{?}n\right)?\mathrm{?}?\left(n\mathrm{?}c\right),\hfill \\ \hfill c_t+u?\mathrm{?}c& \hfill =\hfill & \mathrm{\Delta }c?nc,\hfill \\ \hfill u_t+\mathrm{?}P& \hfill =\hfill & \mathrm{\Delta }u+n\mathrm{?}?,\mathrm{?}?u=0,\hfill \end{array}$

is considered in bounded convex three-dimensional domains, where $?\in W^{2,\infty }(\mathrm{\Omega })$ is given.The paper develops an analytical approach which consists in a combination of energy-based arguments and maximal Sobolev regularity theory, and which allows for the construction of global bounded weak solutions to an associated initial-boundary value problem under the assumption that

(0.1)$m>\frac{9}{8}.$

Moreover, the obtained solutions are shown to approach the spatially homogeneous steady state $(\frac{1}{|\mathrm{\Omega }|}{\int }_{\mathrm{\Omega }}{n}_0,0,0)$ in the large time limit.This extends previous results which either relied on different and apparently less significant energy-type structures, or on completely alternative approaches, and thereby exclusively achieved comparable results under hypotheses stronger than (0.1).     

W01,1 solutions in some borderline cases of Calderon–Zygmund theory

In this paper we study the existence of $W_0^{1,1}(\Omega )$ distributional solutions of Dirichlet problems whose simplest example is$\{\begin{array}{cc}?\mathrm{div}\left({|\mathrm{?}u|}^{p?2}\mathrm{?}u\right)=f(x),\hfill & \text{in}\Omega ;\hfill \\ u=0,\hfill & \text{on}?\Omega .\hfill \end{array}$     

Homogenization of a Dirichlet semilinear elliptic problem with a strong singularity at u = 0 in a domain with many small holes

In the present paper we perform the homogenization of the semilinear elliptic problem

$\{\begin{array}{cc}{u}^{\epsilon }\ge 0\hfill & \text{in}{\mathrm{\Omega }}^{\epsilon },\hfill \\ ?divA(x)D{u}^{\epsilon }=F(x,u^{\epsilon })\hfill & \text{in}{\mathrm{\Omega }}^{\epsilon },\hfill \\ u^{\epsilon }=0\hfill & \text{on}?{\mathrm{\Omega }}^{\epsilon }.\hfill \end{array}$

In this problem $F(x,s)$ is a Carathéodory function such that $0\le F(x,s)\le h(x)/\mathrm{\Gamma }(s)$ a.e. $x\in \mathrm{\Omega }$ for every $s>0$, with h in some $L^r(\mathrm{\Omega })$ and Γ a $C^1([0,+\infty [)$ function such that $\mathrm{\Gamma }(0)=0$ and ${\mathrm{\Gamma }}^{\prime }(s)>0$ for every $s>0$. On the other hand the open sets ${\mathrm{\Omega }}^{\epsilon }$ are obtained by removing many small holes from a fixed open set Ω in such a way that a “strange term” $\mu u^0$ appears in the limit equation in the case where the function $F(x,s)$ depends only on x.We already treated this problem in the case of a “mild singularity”, namely in the case where the function $F(x,s)$ satisfies $0\le F(x,s)\le h(x)(\frac{1}{s}+1)$. In this case the solution $u^{\epsilon }$ to the problem belongs to $H_0^1({\mathrm{\Omega }}^{\epsilon })$ and its definition is a “natural” and rather usual one.In the general case where $F(x,s)$ exhibits a “strong singularity” at $u=0$, which is the purpose of the present paper, the solution $u^{\epsilon }$ to the problem only belongs to $H_{\text{loc}}^1({\mathrm{\Omega }}^{\epsilon })$ but in general does not belong to $H_0^1({\mathrm{\Omega }}^{\epsilon })$ anymore, even if $u^{\epsilon }$ vanishes on $?{\mathrm{\Omega }}^{\epsilon }$ in some sense. Therefore we introduced a new notion of solution (in the spirit of the solutions defined by transposition) for problems with a strong singularity. This definition allowed us to obtain existence, stability and uniqueness results.In the present paper, using this definition, we perform the homogenization of the above semilinear problem and we prove that in the homogenized problem, the “strange term” $\mu u^0$ still appears in the left-hand side while the source term $F(x,u^0)$ is not modified in the right-hand side.     

Global existence and blow-up of solutions for the heat equation with exponential nonlinearity

We are concerned with the existence of global in time solution for a semilinear heat equation with exponential nonlinearity

(P)$\{\begin{array}{cc}{?}_{t}u=\mathrm{\Delta }u+e^u,\hfill & x\in {\mathbf{R}}^N,t>0,\hfill \\ u(x,0)=u_0(x),\hfill & x\in {\mathbf{R}}^N,\hfill \end{array}$

where $u_0$ is a continuous initial function. In this paper, we consider the case where $u_0$ decays to ?∞ at space infinity, and study the optimal decay bound classifying the existence of global in time solutions and blowing up solutions for (P). In particular, we point out that the optimal decay bound for $u_0$ is related to the decay rate of forward self-similar solutions of ${?}_{t}u=\mathrm{\Delta }u+e^u$.     

Nonexistence and symmetry of solutions to some fractional laplacian equations in the upper half space

In this article, we consider the fractional Laplacian equation(-△)~(α/2)u = K(x)f(u), x ∈ R_+~n,u ≡ 0, x/∈R_+~n,where 0 α 2, R_+~n:= {x =(x_1, x_2, ···, x_n)|x n 0}. When K is strictly decreasing with respect to |x′|, the symmetry of positive solutions is proved, where x′=(x_1, x_2, ···, x_(n-1)) ∈R~(n-1). When K is strictly increasing with respect to x n or only depend on x n, the nonexistence of positive solutions is obtained.     

Topological bounds for Fourier coefficients and applications to torsion

Let $\mathrm{\Omega }?{\mathbb{R}}^2$ be a bounded convex domain in the plane and consider

$\begin{array}{cc}\hfill ?\mathrm{\Delta }u& =1\text{in}\mathrm{\Omega }\hfill \\ \hfill u& =0\text{on}?\mathrm{\Omega }.\hfill \end{array}$

If u assumes its maximum in $x_0\in \mathrm{\Omega }$, then the eccentricity of level sets close to the maximum is determined by the Hessian $D^{2}u(x_0)$. We prove that $D^{2}u(x_0)$ is negative definite and give a quantitative bound on the spectral gap

$\begin{array}{c}{\lambda }_{\max }(D^{2}u(x_0))\le ?c_1\exp ?(?c_2\frac{\mathrm{diam}(\mathrm{\Omega })}{\mathrm{inrad}(\mathrm{\Omega })})\hfill \\ \text{for universal}{c}_1,c_2>0.\hfill \end{array}$

This is sharp up to constants. The proof is based on a new lower bound for Fourier coefficients whose proof has a topological component: if $f:\mathbb{T}\to \mathbb{R}$ is continuous and has n sign changes, then

$\sum _{k=0}^{n/2}\left|〈f,\sin ?kx〉\right|+\left|〈f,\cos ?kx〉\right|{?}_n\frac{{|f6}_{L^1(\mathbb{T})}^{n+1}}{{6f6}_{L^{\infty }(\mathbb{T})}^n}.$

This statement immediately implies estimates on higher derivatives of harmonic functions u in the unit ball: if u is very flat in the origin, then the boundary function $u(\cos ?t,\sin ?t):\mathbb{T}\to \mathbb{R}$ has to have either large amplitude or many roots. It also implies that the solution of the heat equation starting with $f:\mathbb{T}\to \mathbb{R}$ cannot decay faster than $～\exp ?(?{(\mathrm{\#}\text{sign changes})}^{2}t/4)$.     

A proof of the Tn conjecture: Centralizers,Jacobians and spectrally arbitrary sign patterns

We prove the so-called $T_n$ conjecture: for every real-monic polynomial $p(x)$ of degree $n?2$ there exists an n by n matrix with sign pattern$T_n=\left[\begin{array}{ccccc}\hfill -\hfill & \hfill +\hfill & \hfill 0\hfill & \hfill ?\hfill & \hfill 0\hfill \\ \hfill -\hfill & \hfill 0\hfill & \hfill ?\hfill & \hfill \hfill & \hfill ?\hfill \\ \hfill 0\hfill & \hfill ?\hfill & \hfill ?\hfill & \hfill ?\hfill & \hfill 0\hfill \\ \hfill ?\hfill & \hfill \hfill & \hfill ?\hfill & \hfill 0\hfill & \hfill +\hfill \\ \hfill 0\hfill & \hfill ?\hfill & \hfill 0\hfill & \hfill -\hfill & \hfill +\hfill \end{array}\right]\text{,}$whose characteristic polynomial is $p(x)$. The proof converts the problem of determining the nonsingularity of a certain Jacobi matrix to the problem of proving the non-existence of a nonzero matrix B that commutes with a nilpotent matrix with sign pattern $T_n$ and has zeros in positions $(1\text{,}1)$, and $(j+1\text{,}j)$ for $j=2\text{,}\dots \text{,}n-1$.     

Construction and stability of blowup solutions for a non-variational semilinear parabolic systemConstruction et stabilité de solutions explosives pour un système parabolique sémilinéaire non-variationel

We consider the following parabolic system whose nonlinearity has no gradient structure:

$\{\begin{array}{cc}{?}_{t}u=\mathrm{\Delta }u+|v{|}^{p?1}v,\hfill & {?}_{t}v=\mu \mathrm{\Delta }v+|u{|}^{q?1}u,\hfill \\ u(?,0)=u_0,\hfill & v(?,0)=v_0,\hfill \end{array}$

in the whole space ${\mathbb{R}}^N$, where $p,q>1$ and $\mu >0$. We show the existence of initial data such that the corresponding solution to this system blows up in finite time $T(u_0,v_0)$ simultaneously in u and v only at one blowup point a, according to the following asymptotic dynamics:

$\{\begin{array}{cc}\hfill & u(x,t)～\mathrm{\Gamma }{[(T?t)(1+\frac{b|x?a{|}^2}{(T?t)|\log ?(T?t)|})]}^{?\frac{(p+1)}{pq?1}},\hfill \\ \hfill & v(x,t)～\gamma {[(T?t)(1+\frac{b|x?a{|}^2}{(T?t)|\log ?(T?t)|})]}^{?\frac{(q+1)}{pq?1}},\hfill \end{array}$

with $b=b(p,q,\mu )>0$ and $(\mathrm{\Gamma },\gamma )=(\mathrm{\Gamma }(p,q),\gamma (p,q))$. The construction relies on the reduction of the problem to a finite dimensional one and a topological argument based on the index theory to conclude. Two major difficulties arise in the proof: the linearized operator around the profile is not self-adjoint even in the case $\mu =1$; and the fact that the case $\mu \ne 1$ breaks any symmetry in the problem. In the last section, through a geometrical interpretation of quantities of blowup parameters whose dimension is equal to the dimension of the finite dimensional problem, we are able to show the stability of these blowup behaviors with respect to perturbations in initial data.     

A semilinear elliptic equation with a mild singularity at u = 0: Existence and homogenization

In this paper we consider singular semilinear elliptic equations whose prototype is the following

$\{\begin{array}{cc}?divA(x)Du=f(x)g(u)+l(x)\hfill & \text{in}\mathrm{\Omega },\hfill \\ u=0\hfill & \text{on}?\mathrm{\Omega },\hfill \end{array}$

where Ω is an open bounded set of ${\mathbb{R}}^N,N\ge 1$, $A\in L^{\infty }{(\mathrm{\Omega })}^{N\times N}$ is a coercive matrix, $g:[0,+\infty [\to [0,+\infty ]$ is continuous, and $0\le g(s)\le \frac{1}{s^{\gamma }}+1$ for every $s>0$, with $0<\gamma \le 1$ and $f,l\in L^r(\mathrm{\Omega })$, $r=\frac{2N}{N+2}$ if $N\ge 3$, $r>1$ if $N=2$, $r=1$ if $N=1$, $f(x),l(x)\ge 0$ a.e. $x\in \mathrm{\Omega }$.We prove the existence of at least one nonnegative solution as well as a stability result; we also prove uniqueness if $g(s)$ is nonincreasing or “almost nonincreasing”.Finally, we study the homogenization of these equations posed in a sequence of domains ${\mathrm{\Omega }}^{\epsilon }$ obtained by removing many small holes from a fixed domain Ω.