## Displaying similar documents to “Separable solutions of quasilinear Lane–Emden equations”

### Multiplicity results for a class of concave-convex elliptic systems involving sign-changing weight functions

Annales Polonici Mathematici

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Our main purpose is to establish the existence of weak solutions of second order quasilinear elliptic systems ⎧ $-{\Delta }_{p}u+{|u|}^{p-2}u={f}_{1\lambda ₁}{\left(x\right)|u|}^{q-2}u+2\alpha /\left(\alpha +\beta \right){g}_{\mu }{|u|}^{\alpha -2}u{|v|}^{\beta }$, x ∈ Ω, ⎨ $-{\Delta }_{p}v+{|v|}^{p-2}v={f}_{2\lambda ₂}{\left(x\right)|v|}^{q-2}v+2\beta /\left(\alpha +\beta \right){g}_{\mu }{|u|}^{\alpha }{|v|}^{\beta -2}v$, x ∈ Ω, ⎩ u = v = 0, x∈ ∂Ω, where 1 < q < p < N and $\Omega \subset {ℝ}^{N}$ is an open bounded smooth domain. Here λ₁, λ₂, μ ≥ 0 and ${f}_{i{\lambda }_{i}}\left(x\right)={\lambda }_{i}{f}_{i+}\left(x\right)+{f}_{i-}\left(x\right)$ (i = 1,2) are sign-changing functions, where ${f}_{i±}\left(x\right)=max±{f}_{i}\left(x\right),0$, ${g}_{\mu }\left(x\right)=a\left(x\right)+\mu b\left(x\right)$, and ${\Delta }_{p}u={div\left(|\nabla u|}^{p-2}\nabla u\right)$ denotes the p-Laplace operator. We use variational methods.

### Existence and nonexistence of solutions for a quasilinear elliptic system

Annales Polonici Mathematici

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By a sub-super solution argument, we study the existence of positive solutions for the system ⎧$-{\Delta }_{p}u=a₁\left(x\right)F₁\left(x,u,v\right)$ in Ω, ⎪$-{\Delta }_{q}v=a₂\left(x\right)F₂\left(x,u,v\right)$ in Ω, ⎨u,v > 0 in Ω, ⎩u = v = 0 on ∂Ω, where Ω is a bounded domain in ${ℝ}^{N}$ with smooth boundary or $\Omega ={ℝ}^{N}$. A nonexistence result is obtained for radially symmetric solutions.

### Energy and Morse index of solutions of Yamabe type problems on thin annuli

Journal of the European Mathematical Society

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We consider the Yamabe type family of problems $\left({P}_{\epsilon }\right):-\Delta {u}_{\epsilon }={u}_{\epsilon }^{\left(n+2\right)/\left(n-2\right)}$, ${u}_{\epsilon }>0$ in ${A}_{\epsilon }$, ${u}_{\epsilon }=0$ on $\partial {A}_{\epsilon }$, where ${A}_{\epsilon }$ is an annulus-shaped domain of ${ℝ}^{n}$, $n\ge 3$, which becomes thinner as $\epsilon \to 0$. We show that for every solution ${u}_{\epsilon }$, the energy ${\int }_{{A}_{\epsilon }}|\nabla {u}_{|}^{2}$ as well as the Morse index tend to infinity as $\epsilon \to 0$. This is proved through a fine blow up analysis of appropriate scalings of solutions whose limiting profiles are regular, as well as of singular solutions of some elliptic problem on ${ℝ}^{n}$, a half-space or an infinite strip. Our argument also involves a Liouville...

### Recent results on stationary critical Kirchhoff systems in closed manifolds

Séminaire Laurent Schwartz — EDP et applications

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We report on results we recently obtained in Hebey and Thizy [11, 12] for critical stationary Kirchhoff systems in closed manifolds. Let $\left({M}^{n},g\right)$ be a closed $n$-manifold, $n\ge 3$. The critical Kirchhoff systems we consider are written as $\left(a+b\sum _{j=1}^{p}{\int }_{M}{|\nabla {u}_{j}|}^{2}d{v}_{g}\right){\Delta }_{g}{u}_{i}+\sum _{j=1}^{p}{A}_{ij}{u}_{j}={\left|U\right|}^{{2}^{☆}-2}{u}_{i}$ for all $i=1,\cdots ,p$, where ${\Delta }_{g}$ is the Laplace-Beltrami operator, $A$ is a ${C}^{1}$-map from $M$ into the space ${M}_{s}^{p}\left(ℝ\right)$ of symmetric $p×p$ matrices with real entries, the ${A}_{ij}$’s are the components of $A$, $U=\left({u}_{1},\cdots ,{u}_{p}\right)$, $|U|:M\to ℝ$ is the Euclidean norm of $U$, ${2}^{☆}=\frac{2n}{n-2}$ is the critical Sobolev exponent, and...

### Arbitrary number of positive solutions for an elliptic problem with critical nonlinearity

Journal of the European Mathematical Society

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We show that the critical nonlinear elliptic Neumann problem $\Delta u-\mu u+{u}^{7/3}=0$ in $\Omega$, $u>0$ in $\Omega$, $\frac{\partial u}{\partial \nu }=0$ on $\partial \Omega$, where $\Omega$ is a bounded and smooth domain in ${ℝ}^{5}$, has arbitrarily many solutions, provided that $\mu >0$ is small enough. More precisely, for any positive integer $K$, there exists ${\mu }_{K}>0$ such that for $0<\mu <{\mu }_{K}$, the above problem has a nontrivial solution which blows up at $K$ interior points in $\Omega$, as $\mu \to 0$. The location of the blow-up points is related to the domain geometry. The solutions are obtained as critical points of some finite-dimensional...

### Invariance of the parity conjecture for $p$-Selmer groups of elliptic curves in a ${D}_{2{p}^{n}}$-extension

Bulletin de la Société Mathématique de France

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We show a $p$-parity result in a ${D}_{2{p}^{n}}$-extension of number fields $L/K$ ($p\ge 5$) for the twist $1\oplus \eta \oplus \tau$: $W\left(E/K,1\oplus \eta \oplus \tau \right)={\left(-1\right)}^{〈1\oplus \eta \oplus \tau ,{X}_{p}\left(E/L\right)〉}$, where $E$ is an elliptic curve over $K$, $\eta$ and $\tau$ are respectively the quadratic character and an irreductible representation of degree $2$ of $\mathrm{Gal}\left(L/K\right)={D}_{2{p}^{n}}$, and ${X}_{p}\left(E/L\right)$ is the $p$-Selmer group. The main novelty is that we use a congruence result between ${\epsilon }_{0}$-factors (due to Deligne) for the determination of local root numbers in bad cases (places of additive reduction above 2 and 3). We also give applications to the $p$-parity conjecture...

### On the potential theory of some systems of coupled PDEs

Commentationes Mathematicae Universitatis Carolinae

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In this paper we study some potential theoretical properties of solutions and super-solutions of some PDE systems (S) of type ${L}_{1}u=-{\mu }_{1}v$, ${L}_{2}v=-{\mu }_{2}u$, on a domain $D$ of ${ℝ}^{d}$, where ${\mu }_{1}$ and ${\mu }_{2}$ are suitable measures on $D$, and ${L}_{1}$, ${L}_{2}$ are two second order linear differential elliptic operators on $D$ with coefficients of class ${𝒞}^{\infty }$. We also obtain the integral representation of the nonnegative solutions and supersolutions of the system (S) by means of the Green kernels and Martin boundaries associated with ${L}_{1}$ and ${L}_{2}$, and...

### A bifurcation theory for some nonlinear elliptic equations

Colloquium Mathematicae

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We deal with the problem ⎧ -Δu = f(x,u) + λg(x,u), in Ω, ⎨ (${P}_{\lambda }$) ⎩ ${u}_{\mid \partial \Omega }=0$ where Ω ⊂ ℝⁿ is a bounded domain, λ ∈ ℝ, and f,g: Ω×ℝ → ℝ are two Carathéodory functions with f(x,0) = g(x,0) = 0. Under suitable assumptions, we prove that there exists λ* > 0 such that, for each λ ∈ (0,λ*), problem (${P}_{\lambda }$) admits a non-zero, non-negative strong solution ${u}_{\lambda }\in {\bigcap }_{p\ge 2}{W}^{2,p}\left(\Omega \right)$ such that $li{m}_{\lambda \to 0⁺}||{u}_{\lambda }{||}_{{W}^{2,p}\left(\Omega \right)}=0$ for all p ≥ 2. Moreover, the function $\lambda ↦{I}_{\lambda }\left({u}_{\lambda }\right)$ is negative and decreasing in ]0,λ*[, where ${I}_{\lambda }$ is the energy functional related to (${P}_{\lambda }$). ...

### Beyond two criteria for supersingularity: coefficients of division polynomials

Journal de Théorie des Nombres de Bordeaux

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Let $f\left(x\right)$ be a cubic, monic and separable polynomial over a field of characteristic $p\ge 3$ and let $E$ be the elliptic curve given by ${y}^{2}=f\left(x\right)$. In this paper we prove that the coefficient at ${x}^{\frac{1}{2}p\left(p-1\right)}$ in the $p$–th division polynomial of $E$ equals the coefficient at ${x}^{p-1}$ in $f{\left(x\right)}^{\frac{1}{2}\left(p-1\right)}$. For elliptic curves over a finite field of characteristic $p$, the first coefficient is zero if and only if $E$ is supersingular, which by a classical criterion of Deuring (1941) is also equivalent to the vanishing of the second coefficient. So the...

### Positivity and anti-maximum principles for elliptic operators with mixed boundary conditions

Journal of the European Mathematical Society

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We consider linear elliptic equations $-\Delta u+q\left(x\right)u=\lambda u+f$ in bounded Lipschitz domains $D\subset {ℝ}^{N}$ with mixed boundary conditions $\partial u/\partial n=\sigma \left(x\right)\lambda u+g$ on $\partial D$. The main feature of this boundary value problem is the appearance of $\lambda$ both in the equation and in the boundary condition. In general we make no assumption on the sign of the coefficient $\sigma \left(x\right)$. We study positivity principles and anti-maximum principles. One of our main results states that if $\sigma$ is somewhere negative, $q\ge 0$ and ${\int }_{D}q\left(x\right)dx>0$ then there exist two eigenvalues ${\lambda }_{-1}$, ${\lambda }_{1}$ such the positivity...

### Capacitary estimates of positive solutions of semilinear elliptic equations with absorbtion

Journal of the European Mathematical Society

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Let $\Omega$ be a bounded domain of class ${C}^{2}$ in $ℝ$N and let $K$ be a compact subset of $\partial \Omega$. Assume that $q\ge \left(N+1\right)/\left(N-1\right)$ and denote by ${U}_{K}$ the maximal solution of $-\Delta u+{u}^{q}=0$ in $\Omega$ which vanishes on $\partial \Omega \setminus K$. We obtain sharp upper and lower estimates for ${U}_{K}$ in terms of the Bessel capacity ${C}_{2/q,{q}^{\text{'}}}$ and prove that ${U}_{K}$ is $\sigma$-moderate. In addition we describe the precise asymptotic behavior of ${U}_{K}$ at points $\sigma \in K$, which depends on the “density” of $K$ at $\sigma$, measured in terms of the capacity ${C}_{2/q,{q}^{\text{'}}}$.

### On square functions associated to sectorial operators

Bulletin de la Société Mathématique de France

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We give new results on square functions ${\parallel x\parallel }_{F}=\parallel {\left({\int }_{0}^{\infty }{\left|F\left(tA\right)x\right|}^{2}\frac{\phantom{\rule{0.55542pt}{0ex}}\mathrm{d}t}{t}\right)}^{1/2}{\parallel }_{p}$ associated to a sectorial operator $A$ on ${L}^{p}$ for $1<p<\infty$. Under the assumption that $A$ is actually $R$-sectorial, we prove equivalences of the form ${K}^{-1}{\parallel x\parallel }_{G}\le {\parallel x\parallel }_{F}\le K{\parallel x\parallel }_{G}$ for suitable functions $F,G$. We also show that $A$ has a bounded ${H}^{\infty }$ functional calculus with respect to ${\parallel \phantom{\rule{0.166667em}{0ex}}.\phantom{\rule{0.166667em}{0ex}}\parallel }_{F}$. Then we apply our results to the study of conditions under which we have an estimate $\parallel {\left({\int }_{0}^{\infty }|C{\mathrm{e}}^{-tA}{\left(x\right)|}^{2}\mathrm{d}t{\right)}^{1/2}\parallel }_{q}\le M{\parallel x\parallel }_{p}$, when $-A$ generates a bounded semigroup ${\mathrm{e}}^{-tA}$ on ${L}^{p}$ and $C:D\left(A\right)\to {L}^{q}$ is a linear mapping.

### On annealed elliptic Green's function estimates

Mathematica Bohemica

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We consider a random, uniformly elliptic coefficient field $a$ on the lattice ${ℤ}^{d}$. The distribution $〈·〉$ of the coefficient field is assumed to be stationary. Delmotte and Deuschel showed that the gradient and second mixed derivative of the parabolic Green’s function $G\left(t,x,y\right)$ satisfy optimal annealed estimates which are ${L}^{2}$ and ${L}^{1}$, respectively, in probability, i.e., they obtained bounds on $〈|{\nabla }_{x}G\left(t,x,y\right){{|}^{2}〉}^{1/2}$ and $〈|{\nabla }_{x}{\nabla }_{y}G\left(t,x,y\right)|〉$. In particular, the elliptic Green’s function $G\left(x,y\right)$ satisfies optimal annealed bounds. In their recent work,...

### On the topology of polynomials with bounded integer coefficients

Journal of the European Mathematical Society

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For a real number $q>1$ and a positive integer $m$, let ${Y}_{m}\left(q\right):=\left\{{\sum }_{i=0}^{n}{ϵ}_{i}{q}^{i}:{ϵ}_{i}\in \left\{0,±1,...,±m\right\},n=0,1,...\right\}$. In this paper, we show that ${Y}_{m}\left(q\right)$ is dense in $ℝ$ if and only if $q and $q$ is not a Pisot number. This completes several previous results and answers an open question raised by Erdös, Joó and Komornik .

### Perturbed nonlinear degenerate problems in ${ℝ}^{N}$

Applicationes Mathematicae

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Via critical point theory we establish the existence and regularity of solutions for the quasilinear elliptic problem ⎧ $div\left(x,\nabla u\right)+{a\left(x\right)|u|}^{p-2}u={g\left(x\right)|u|}^{p-2}u+h\left(x\right){|u|}^{s-1}u$ in ${ℝ}^{N}$ ⎨ ⎩ u > 0, $li{m}_{|x|\to \infty }u\left(x\right)=0$, where 1 < p < N; a(x) is assumed to satisfy a coercivity condition; h(x) and g(x) are not necessarily bounded but satisfy some integrability restrictions.

### ${𝒞}^{k}$-regularity for the $\overline{\partial }$-equation with a support condition

Czechoslovak Mathematical Journal

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Let $D$ be a ${𝒞}^{d}$ $q$-convex intersection, $d\ge 2$, $0\le q\le n-1$, in a complex manifold $X$ of complex dimension $n$, $n\ge 2$, and let $E$ be a holomorphic vector bundle of rank $N$ over $X$. In this paper, ${𝒞}^{k}$-estimates, $k=2,3,\cdots ,\infty$, for solutions to the $\overline{\partial }$-equation with small loss of smoothness are obtained for $E$-valued $\left(0,s\right)$-forms on $D$ when $n-q\le s\le n$. In addition, we solve the $\overline{\partial }$-equation with a support condition in ${𝒞}^{k}$-spaces. More precisely, we prove that for a $\overline{\partial }$-closed form $f$ in ${𝒞}_{0,q}^{k}\left(X\setminus D,E\right)$, $1\le q\le n-2$, $n\ge 3$, with compact support and for $\epsilon$ with $0<\epsilon <1$ there...

### Sum-product theorems and incidence geometry

Journal of the European Mathematical Society

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In this paper we prove the following theorems in incidence geometry. 1. There is $\delta >0$ such that for any ${P}_{1},\cdots ,{P}_{4}$, and ${Q}_{1},\cdots ,{Q}_{n}\in {ℂ}^{2}$, if there are $\le {n}^{\left(1+\delta \right)/2}$ many distinct lines between ${P}_{i}$ and ${Q}_{j}$ for all $i$, $j$, then ${P}_{1},\cdots ,{P}_{4}$ are collinear. If the number of the distinct lines is $ then the cross ratio of the four points is algebraic. 2. Given $c>0$, there is $\delta >0$ such that for any ${P}_{1},{P}_{2},{P}_{3}\in {ℂ}^{2}$ noncollinear, and ${Q}_{1},\cdots ,{Q}_{n}\in {ℂ}^{2}$, if there are $\le c{n}^{1/2}$ many distinct lines between ${P}_{i}$ and ${Q}_{j}$ for all $i$, $j$, then for any $P\in {ℂ}^{2}\setminus \left\{{P}_{1},{P}_{2},{P}_{3}\right\}$, we have $\delta n$ distinct lines between $P$ and ${Q}_{j}$. 3. Given...