Let $\Omega\subset\mathbb R^n$ be a bounded domain, and let $f$ be a nonnegative, nondecreasing function satisfying the Keller-Osserman condition. We study boundary blow-up solutions of $\Delta u=f(u)$ in $\Omega$. Although existence is classical, uniqueness under these assumptions is known in balls but remains open even for smooth convex domains. We identify the normalized gradient $Q_u=|\nabla u|^2/(2F(u))$, $F'=f$, as a quantity governing uniqueness. Under a structural condition on $f$, a boundary blow-up solution $u$ is unique if $\limsup_{x\to\partial\Omega}Q_u(x)\le 1$, without any regularity assumption on $\partial \Omega$. For $C^{1,1}$ domains, assuming a growth condition on $f$, we prove $Q_u(x)\to 1$ for every boundary blow-up solution and hence obtain uniqueness under the structural condition. For convex domains, we prove $Q_u\le 1$ for the minimal boundary blow-up solution and obtain uniqueness when $\sqrt F$ is eventually convex, without imposing any additional boundary regularity.
We study the uniqueness and boundary behavior of nonzero convex Aleksandrov solutions to $\det D^2 u=M|u|^p\nu$ with zero boundary values on bounded convex domains in $\mathbb{R}^n (n \geq 2)$. For $0<p<n$, we prove the uniqueness of nonzero convex solutions in the finite-energy class when $\nu$ is a locally finite Bor...
Berestycki and Graham proved large-dilation uniqueness for bounded positive solutions of \[ -\Delta u=f(u)\quad\hbox{in }\kappa\Omega, \qquad u+\alpha\partial_\nu u=0\quad\hbox{on }\partial(\kappa\Omega),\] when $\alpha$ is fixed, and remarked that the dilation threshold should not depend on $\alpha$. We show that it d...
We study the Neumann problem $u_t-\varepsilon\Delta u=e^u-1-au\ (a>1)$ on a smooth bounded domain $\Omega\subset\mathbb{R}^2$. For the spatially homogeneous problem, $0$ is stable, the positive equilibrium $\xi_a$ is unstable, and solutions starting above $\xi_a$ blow up in finite time. Although finite-time blow-up per...
We establish interior maximal $L^{q_c}$-regularity for bounded strong solutions of $u_t-\Delta u+|Du|^\gamma=f$ in $\mathbb{T}^d\times(0,T)$, where $d\geq 2$, $\gamma>2$, and $q_c=(d+2)(\gamma-1)/\gamma$. The estimates are uniform for uniformly bounded families of solutions whose source terms range over a bounded, unif...
Let $D\subset\mathbb C^n$, $n\geq 2$, be a bounded pseudoconvex domain with smooth boundary, and let $H^p_\omega(D)$ be the Hardy space defined using a weighted boundary measure $\omega\,d\sigma$, where $\omega$ is bounded above and bounded away from zero. For every $0<p<\infty$, $p\neq2$, we prove that each surjective...
Ren-Yu Chen, Song-Ying Li, Su-Juan Long et al.· 0 citations
Let $\Omega\subset\mathbb R^2$ be a smooth bounded domain containing the origin and invariant under reflection across the coordinate axes, and let $0<\lambda<\lambda_1(\Omega)$, where $\Lambda_1 (\Omega)$ is the first eigenvalue for $-\Delta$ on $\Omega$ under Dirichlet boundary conditions. For every fixed integer $k\g...
M. del Pino, Ignacio A. Guerra, M. Musso· 0 citations
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