We study the leading order behaviour of positive solutions of the equation $-\Delta u+ϵu-{\left|u\right|}^{p-2}u+{\left|u\right|}^{q-2}u=0,x\in {ℝ}^N$, where $N\ge 3$, $q>p>2$ and when $ϵ>0$ is a small parameter. We give a complete characterization of all possible asymptotic regimes as a function of $p$, $q$ and $N$. The behavior of solutions depends sensitively on whether $p$ is less, equal or bigger than the critical Sobolev exponent $2^{*}=\frac{2N}{N-2}$. For $p<2^{*}$ the solution asymptotically coincides with the solution of the equation in which the last term is absent. For $p>2^{*}$ the solution asymptotically...

On the unit disk $B_1\subset {ℝ}^2$ we study the Moser-Trudinger functional $E\left(u\right)={\int }_{B_1}\left(e^{u^2}-1\right)dx,u\in H_0^1\left(B_1\right)$ and its restrictions ${E|}_{M_{\Lambda }}$, where $M_{\Lambda }:=\{u\in H_0^1\left(B_1\right){:\parallel u\parallel }_{H_0^1}^2=\Lambda \}$ for $\Lambda >0$. We prove that if a sequence $u_k$ of positive critical points of ${E|}_{M_{{\Lambda }_k}}$ (for some ${\Lambda }_k>0$) blows up as $k\to \infty$, then ${\Lambda }_k\to 4\pi$, and $u_k\to 0$ weakly in $H_0^1\left(B_1\right)$ and strongly in $C_{\mathrm{loc}}^1({\overline{B}}_1\setminus \left\{0\right\})$. Using this fact we also prove that when $\Lambda$ is large enough, then ${E|}_{M_{\Lambda }}$ has no positive critical point, complementing previous existence results by Carleson-Chang, M. Struwe and Lamm-Robert-Struwe.

Let $𝔻$ denote the unit disk $\{z:|z|<1\}$ in the complex plane $ℂ$. In this paper, we study a family of polynomials $P$ with only one zero lying outside $\overline{𝔻}$.  We establish  criteria for $P$ to satisfy implying that each of $P$ and $P^{\text{'}}$  has exactly one critical point outside $\overline{𝔻}$.

The role of the second critical exponent $p=(n+1)/(n-3)$, the Sobolev critical exponent in one dimension less, is investigated for the classical Lane–Emden–Fowler problem $\Delta u+u^p=0$, $u>0$ under zero Dirichlet boundary conditions, in a domain $\Omega$ in ${ℝ}^n$ with bounded, smooth boundary. Given $\Gamma$, a geodesic of the boundary with negative inner normal curvature we find that for $p=(n+1)/(n-3-\epsilon )$, there exists a solution $u_{\epsilon }$ such that $|\nabla u_{\epsilon }{|}^2$ converges weakly to a Dirac measure on $\Gamma$ as $\epsilon \to 0^{+}$, provided that $\Gamma$ is nondegenerate in the sense of second...

We report on results we recently obtained in Hebey and Thizy [11, 12] for critical stationary Kirchhoff systems in closed manifolds. Let $(M^n,g)$ be a closed $n$-manifold, $n\ge 3$. The critical Kirchhoff systems we consider are written as $$\left(a+b\sum _{j=1}^p{\int }_M{\left|\nabla u_j\right|}^{2}d{v}_g\right){\Delta }_g{u}_i+\sum _{j=1}^p{A}_{ij}{u}_j={\left|U\right|}^{2^{☆}-2}{u}_i$$ 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 $p\times p$ matrices with real entries, the $A_{ij}$’s are the components of $A$, $U=(u_1,\cdots ,u_p)$, $\left|U\right|:M\to ℝ$ is the Euclidean norm of $U$, $2^{☆}=\frac{2n}{n-2}$ is the critical Sobolev exponent, and...

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...

We consider the Yamabe type family of problems $\left(P_{\epsilon }\right):-\Delta u_{\epsilon }=u_{\epsilon }^{(n+2)/\left(n-2\right)}$, $u_{\epsilon }>0$ 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...

For a Morse function $f$ on a compact oriented manifold $M$, we show that $f$ has more critical points than the number required by the Morse inequalities if and only if there exists a certain class of link in $M$ whose components have nontrivial linking number, such that the minimal value of $f$ on one of the components is larger than its maximal value on the other. Indeed we characterize the precise number of critical points of $f$ in terms of the Betti numbers of $M$ and the behavior of $f$ with respect...

For $n=2m\ge 4$, let $\Omega \in {ℝ}^n$ be a bounded smooth domain and $𝒩\subset {ℝ}^L$ a compact smooth Riemannian manifold without boundary. Suppose that $\left\{u_k\right\}\in W^{m,2}(\Omega ,𝒩)$ is a sequence of weak solutions in the critical dimension to the perturbed $m$-polyharmonic maps $$\frac{\mathrm{d}}{\mathrm{d}t}{|}_{t=0}{E}_m\left(\Pi (u+t\xi )\right)=0$$ with ${\Phi }_k\to 0$ in ${\left(W^{m,2}(\Omega ,𝒩)\right)}^{*}$ and $u_k\rightharpoonup u$ weakly in $W^{m,2}(\Omega ,𝒩)$. Then $u$ is an $m$-polyharmonic map. In particular, the space of $m$-polyharmonic maps is sequentially compact for the weak-$W^{m,2}$ topology.

We consider a singularly perturbed elliptic equation ${ϵ}^2\Delta u-V\left(x\right)u+f\left(u\right)=0,u\left(x\right)>0$ on ${ℝ}^N$, ${\mathrm{𝚕𝚒𝚖}}_{\left|x\right|\to \infty }u\left(x\right)=0$, where $V\left(x\right)>0$ for any $x\in {ℝ}^N$. The singularly perturbed problem has corresponding limiting problems $\Delta U-cU+f\left(U\right)=0,U\left(x\right)>0$ on ${ℝ}^N$, ${\mathrm{𝚕𝚒𝚖}}_{\left|x\right|\to \infty }U\left(x\right)=0,c>0$. Berestycki-Lions found almost necessary and sufficient conditions on nonlinearity $f$ for existence of a solution of the limiting problem. There have been endeavors to construct solutions of the singularly perturbed problem concentrating around structurally stable critical points of potential $V$ under possibly general conditions...

Let $\Omega \subset {ℝ}^n$ be a bounded starshaped domain and consider the $(p,q)$-Laplacian problem $$\begin{array}{c}\hfill -{\Delta }_{p}u-{\Delta }_{q}u=\lambda \left(𝐱\right){\left|u\right|}^{p^{☆}-2}u+\mu {\left|u\right|}^{r-2}u\end{array}$$ where $\mu$ is a positive parameter, $1<q\le p<n$, $r\ge p^{☆}$ and $p^{☆}:=\frac{np}{n-p}$ is the critical Sobolev exponent. In this short note we address the question of non-existence for non-trivial solutions to the $(p,q)$-Laplacian problem. In particular we show the non-existence of non-trivial solutions to the problem by using a method based on Pohozaev identity.

We give a simple proof that critical values of any Artin $L$-function attached to a representation $\rho$ with character ${\chi }_{\rho }$ are stable under twisting by a totally even character $\chi$, up to the $dim\rho$-th power of the Gauss sum related to $\chi$ and an element in the field generated by the values of ${\chi }_{\rho }$ and $\chi$ over $\mathbb{Q}$. This extends a result of Coates and Lichtenbaum as well as the previous work of Ward.

Let $C$ be a hyperelliptic curve of genus $g\ge 1$ over a number field $K$ with good reduction outside a finite set of places $S$ of $K$. We prove that $C$ has a Weierstrass model over the ring of integers of $K$ with height effectively bounded only in terms of $g$, $S$ and $K$. In particular, we obtain that for any given number field $K$, finite set of places $S$ of $K$ and integer $g\ge 1$ one can in principle determine the set of $K$-isomorphism classes of hyperelliptic curves over $K$ of genus $g$ with good reduction outside...

We prove that two $C^3$ critical circle maps with the same rotation number in a special set $𝔸$ are $C^{1+\alpha }$ conjugate for some $\alpha >0$ provided their successive renormalizations converge together at an exponential rate in the $C^0$ sense. The set $𝔸$ has full Lebesgue measure and contains all rotation numbers of bounded type. By contrast, we also give examples of $C^{\infty }$ critical circle maps with the same rotation number that are not $C^{1+\beta }$ conjugate for any $\beta >0$. The class of rotation numbers for which such examples exist...

For a smooth curve $\Gamma$ and a set $\Lambda$ in the plane ${ℝ}^2$, let $AC(\Gamma ;\Lambda )$ be the space of finite Borel measures in the plane supported on $\Gamma$, absolutely continuous with respect to the arc length and whose Fourier transform vanishes on $\Lambda$. Following [12], we say that $(\Gamma ,\Lambda )$ is a Heisenberg uniqueness pair if $AC(\Gamma ;\Lambda )=\left\{0\right\}$. In the context of a hyperbola $\Gamma$, the study of Heisenberg uniqueness pairs is the same as looking for uniqueness sets $\Lambda$ of a collection of solutions to the Klein-Gordon equation. In this work, we mainly...

For each n ≥ 1, let $v_{n,k},k\ge 1$ and $u_{n,k},k\ge 1$ be mutually independent sequences of nonnegative random variables and let each of them consist of mutually independent and identically distributed random variables with means v̅ₙ and u̅̅ₙ, respectively. Let $X{ₙ}^B\left(t\right)=(1/cₙ){\sum }_{j=1}^{\left[nt\right]}(v_{n,j}-v̅ₙ)$, $X{ₙ}^A\left(t\right)=(1/cₙ){\sum }_{j=1}^{\left[nt\right]}(u_{n,j}-u̅̅ₙ)$, t ≥ 0, and $Xₙ=X{ₙ}^B-X{ₙ}^A$. The main result gives conditions under which the weak convergence $Xₙ\stackrel{}{\to }X$, where X is a Lévy process, implies $X{ₙ}^B\stackrel{}{\to }{X}^B$ and $X{ₙ}^A\stackrel{}{\to }{X}^A$, where $X^B$ and $X^A$ are mutually independent Lévy processes and $X=X^B-X^A$.

We study the existence of positive solutions of the quasilinear problem ⎧ $-{\Delta }_{N}u+{V\left(x\right)\left|u\right|}^{N-2}u={f(u,|\nabla u|}^{N-2}\nabla u)$, $x\in {ℝ}^N$, ⎨ ⎩ u(x) > 0, $x\in {ℝ}^N$, where ${\Delta }_{N}u={div\left(\right|\nabla u|}^{N-2}\nabla u)$ is the N-Laplacian operator, $V:{ℝ}^N\to ℝ$ is a continuous potential, $f:ℝ\times {ℝ}^N\to ℝ$ is a continuous function. The main result follows from an iterative method based on Mountain Pass techniques.