arXiv subjects
Publications and source records attributed to Giusi Vaira.
We show that the classical Brezis-Nirenberg problem $$ -Δu=u|u| + λu\ \hbox{in}\ Ω, u=0\ \hbox{on}\ \partialΩ, $$ when $Ω$ is a bounded domain in $\mathbb R^6$ has a sign-changing solution which blows-up at a point in $Ω$ as $λ$ approaches a suitable value $λ_0>0.$
We prove the non-degeneracy of the extremals of the Sobolev inequality $$\int\limits_{\mathbb R^N}|\nabla u|^pdx\ge \mathcal S_p\int\limits_{\mathbb R^N}|u|^{Np\over N-p}dx,\ u\in \mathcal D^{1,p}(\mathbb R^N)$$ when $1<p<N,$ as solutions of a critical quasilinear equation involving the $p-$Laplacian.
We prove the existence of solutions $(λ, v)\in \mathbb{R}\times H^{1}(Ω)$ of the elliptic problem \[ \begin{cases} -Δv+(V(x)+λ) v =v^{p}\ &\text{ in $ Ω, $} \ v>0,\qquad \int_Ωv^2\,dx =ρ. \end{cases} \] Any $v$ solving such problem (for some $λ$) is called a normalized solution, where the normalization is settled in $L^2(Ω)$. Here $Ω$ is either the whole space $\mathbb R^N$ or a bounded smooth domain of $\mathbb R^N$, in which case we assume $V\equiv0$ and homogeneous Dirichlet or Neumann boundary conditions. Moreover, $1 1$ if $N=1,2$. Normalized solutions appear in different contexts, such as the study of the Nonlinear Schrödinger equation, or that of quadratic ergodic Mean Field Games systems. We prove the existence of solutions concentrating at suitable points of $Ω$ as the prescribed mass $ρ$ is either small (when $p<1+\frac 4N$) or large (when $p>1+\frac 4N$) or it approaches some critical threshold (when $p=1+\frac 4N$).
We consider weak positive solutions to the critical $p$-Laplace equation with Hardy potential in $\mathbb R^N$ $$-Δ_p u -\fracγ{|x|^p} u^{p-1}=u^{p^*-1}$$ where $1<p<N$, $0\le γ<\left(\frac{N-p}{p}\right)^p$ and $p^*=\frac{Np}{N-p}$. The main result is to show that all the solutions in $\mathcal D^{1, p}(\mathbb R^N)$ are radial and radially decreasing about the origin.
We consider equations of the form $Δu +λ^2 V(x)e^{\,u}=ρ$ in various two dimensional settings. We assume that $V>0$ is a given function, $λ>0$ is a small parameter and $ρ=\mathcal O(1)$ or $ρ\to +\infty$ as $λ\to 0$. In a recent paper we prove the existence of the maximal solutions for a particular choice $V\equiv 1$, $ρ=0$ when the problem is posed in doubly connected domains under Dirichlet boundary conditions. We related the maximal solutions with a novel free boundary problem. The purpose of this note is to derive the corresponding free boundary problems in other settings. Solvability of such problems is, viewed formally, the necessary condition for the existence of the maximal solution.
In this paper we consider the Liouville equation $Δu +λ^2 e^{\,u}=0$ with Dirichlet boundary conditions in a two dimensional, doubly connected domain $Ω$. We show that there exists a simple, closed curve $γ\subset Ω$ such that for a sequence $λ_n\to 0$ and a sequence of solutions $u_{n}$ it holds $\frac{u_{n}}{\log\frac{1}{λ_n}}\to H$, where $H$ is a harmonic function in $Ω\setminusγ$ and $\frac{λ_n^2}{\log\frac{1}{λ_n}}\int_Ωe^{\,u_n}\,dx\to 8πc_Ω$, where $c_Ω$ is a constant depending on the conformal class of $Ω$ only.
We consider the following perturbed critical Dirichlet problem involving the Hardy-Schrödinger operator on a smooth bounded domain $Ω\subset \mathbb{R}^N$, $N\geq 3$, with $0 \in Ω$: $$ \left\{ \begin{array}{ll}-Δu-γ\frac{u}{|x|^2}-εu=|u|^{\frac{4}{N-2}}u &\hbox{in }Ωu=0 & \hbox{on }\partial Ω, \end{array}\right. $$ when $ε>0$ is small and $γ< {(N-2)^2\over4}$. Setting $ γ_j= \frac{(N-2)^2}{4}\left(1-\frac{j(N-2+j)}{N-1}\right)\in(-\infty,0]$ for $j \in \mathbb{N},$ we show that if $γ\leq \frac{(N-2)^2}{4}-1$ and $γ\neq γ_j$ for any $j$, then for small $ε$, the above equation has a positive --non variational-- solution that develops a bubble at the origin. If moreover $γ<\frac{(N-2)^2}{4}-4,$ then for any integer $k \geq 2$, the equation has for small enough $ε$, a sign-changing solution that develops into a superposition of $k$ bubbles with alternating sign centered at the origin. The above result is optimal in the radial case, where the condition that $γ\neq γ_j$ is not necessary. Indeed, it is known that, if $γ> \frac{(N-2)^2}{4}-1$ and $Ω$ is a ball $B$, then there is no radial positive solution for $ε>0$ small. We complete the picture here by showing that, if $γ\geq \frac{(N-2)^2}{4}-4$, then the above problem has no radial sign-changing solutions for $ε>0$ small. These results recover and improve what is known in the non-singular case, i.e., when $γ=0$.
In this paper we study a system which is equivalent to a nonlocal version of the well known Brezis Nirenberg problem. The difficulties related with the lack of compactness are here emphasized by the nonlocal nature of the critical nonlinear term. We prove existence and nonexistence results of positive solutions when $N=3$ and existence of solutions in both the resonance and the nonresonance case for higher dimensions.
We consider the boundary value problem $-Δu + u =λe^u$ in $Ω$ with Neumann boundary condition, where $Ω$ is a bounded smooth domain in $\mathbb R^2$, $λ>0.$ This problem is equivalent to the stationary Keller-Segel system from chemotaxis. We establish the existence of a solution $u_λ$ which exhibits a sharp boundary layer along the entire boundary $\partialΩ$ as $λ\to 0$. These solutions have large mass in the sense that $ \int_Ωλe^{u_λ} \sim |\logλ|.$
Let $(M,g)$ be a compact smooth connected Riemannian manifold (without boundary) of dimension $N\ge7$. Assume $M$ is symmetric with respect to a point $ξ_0$ with non-vanishing Weyl's tensor. We consider the linear perturbation of the Yamabe problem $$(P_ε)\qquad-\mathcal L_g u+εu=u^{N+2\over N-2}\ \hbox{in}\ (M,g) .$$ We prove that for any $k\in \mathbb N$, there exists $ε_k>0$ such that for all $ε\in (0, ε_k)$ the problem $(P_ε)$ has a symmetric solution $u_ε,$ which looks like the superposition of $k$ positive bubbles centered at the point $ξ_0$ as $ε\to 0$. In particular, $ξ_0$ is a {\em towering} blow-up point.
Let $(M,g)$ be a non-locally conformally flat compact Riemannian manifold with dimension $N\ge7.$ We are interested in finding positive solutions to the linear perturbation of the Yamabe problem $$-\mathcal L_g u+εu=u^{N+2\over N-2}\ \hbox{in}\ (M,g) $$ where the first eigenvalue of the conformal laplacian $-\mathcal L_g $ is positive and $ε$ is a small positive parameter. We prove that for any point $ξ_0\in M$ which is non-degenerate and non-vanishing minimum point of the Weyl's tensor and for any integer $k$ there exists a family of solutions developing $k$ peaks collapsing at $ξ_0$ as $ε$ goes to zero. In particular, $ξ_0$ is a non-isolated blow-up point.
We consider the problem of finding pairs $(λ; u)$, with $λ> 0$ and $u$ a harmonic function in a three dimensional torus-like domain, satisfying the nonlinear boundary condition $\partial_ν u = λ\sinh u$ on the boundary. This type of boundary condition arises in corrosion modelling (Butler Volmer condition). We prove existence of solutions which concentrate along some geodesics of the boundary as the parameter $λ$ goes to zero.
We study the existence of nonradial sign-changing solutions to the Schroedinger-Poisson system in dimension N>=3. We construct nonradial sign-changing multi-peak solutions whose peaks are displaced in suitable symmetric configurations and collapse to the same point. The proof is based on the Lyapunov-Schmidt reduction.
We consider the Brezis-Nirenberg problem: $$-Δu =λu + |u|^{p-1}u\qquad \mbox{in}\,\, Ω,\quad u=0\,\, \mbox{on}\,\,\ \partialΩ,$$ where $Ω$ is a smooth bounded domain in $\mathbb R^N$, $N\geq 3$, $p=\frac{N+2}{N-2}$ and $λ>0$. In this paper we prove that, if $Ω$ is symmetric and $N=4,5$, there exists a sign-changing solution whose positive part concentrates and blows-up at the center of symmetry of the domain, while the negative part vanishes, as $λ\rightarrow λ_1$, where $λ_1=λ_1(Ω)$ denotes the first eigenvalue of $-Δ$ on $Ω$, with zero Dirichlet boundary condition.
In this paper, we prove that the Brezis-Nirenberg problem -Δu = |u|^{p-1}u+εu in Ω; u=0 on \partial Ωwhere Ωis a symmetric bounded smooth domain in R^N, N\geq 7 and p = (N+2)/(N-2), has a solution with the shape of a tower of two bubbles with alternate signs, centered at the center of symmetry of the domain, for all ε> 0 sufficiently small.
Let $(M,g)$ be a $n-$dimensional compact Riemannian manifold without boundary and $Γ$ be a non degenerate closed geodesic of $(M,g)$. We prove that the supercritical problem $$-Δ_gu+h u=u^{\frac{n+1}{n-3}\pmε},\ u>0,\ \hbox{in}\ (M,g)$$ has a solution that concentrates along $Γ$ as $ε$ goes to zero, provided the function $h$ and the sectional curvatures along $Γ$ satisfy a suitable condition. A connection with the solution of a class of periodic O.D.E.'s with singularity of attractive or repulsive type is established.
Let $(M,g)$ be a $m$-dimensional compact Riemannian manifold without boundary. Assume $κ\in C^2(M)$ is such that $-Δ_g+κ$ is coercive. We prove the existence of a solution to the supercritical problems $$ -Δ_gu+κu= u^p,\ u>0\quad\hbox{in}\ (M,g)\quad\hbox{and}\quad-Δ_gu+κu=λe^u\quad\hbox{in}\ (M,g) $$ which concentrate s along a $(m-1)-$dimensional submanifold of $M$ as $p\to\infty$ and $λ\to0$, respectively, under suitable symmetry assumptions on the manifold $M$.
We consider the stationary Keller-Segel system from chemotaxis in a ball and we show the existence of a solution concentrating at the boundary of the ball.