We investigate positive solutions of semilinear equations driven by uniformly elliptic strictly $2s$-stable L\'evy operators, where $s\in (0,1)$. We first prove that every positive distributional solution of $-Lu=u^p$ in a punctured domain $D\setminus\{0\}$ satisfies $-Lu=u^p+k\delta_0$ in $D$ for some $k\ge0$, and that necessarily $k=0$ whenever $p\ge d/(d-2s)$. We then study the corresponding Dirichlet problem in which the Dirac mass is replaced by a bounded positive measure, and establish the existence of a critical parameter $k_\mu$: below this threshold minimal positive solutions exist, whereas above it the problem admits no solution. In the symmetric case, we further prove multiplicity below the threshold, as well as existence and uniqueness at the threshold itself.
For $p>1$, we consider the boundary value problem for fully nonlinear degenerate elliptic equations $-\lambda_i(D^2u)+|Du|^p+\gamma u=f(x)$ in bounded domains with Dirichlet or boundary blow-up conditions; here $\lambda_i(D^2u)$ denotes the $i$-th eigenvalue of the Hessian. We study existence and nonexistence of soluti...
I. Birindelli, G. Galise, Hitoshi Ishii· 0 citations
In this paper, we study isolated singularities of the following semilinear elliptic equation $-\Delta u+\frac12 x\cdot \nabla u+\frac{1}{q-1}u-u^q=0$ in $\Omega \setminus \{0\}$, where $n\ge 3$, $\Omega \subset \mathbb{R}^n$ is a domain, $0 \in \Omega$ and $q>1$. This equation arises in the study of blow-up profiles of...
We develop a pluripotential approach to complex Hessian equations on compact Hermitian manifolds. In this setting, the lack of closedness of the background metric introduces torsion terms that prevent a direct extension of the K\"ahler theory. Our main result is a uniform $L^\infty$ estimate for bounded $\omega$-$m$-su...
We investigate positive weak solutions of the critical $p$-Laplace equation $$ -\Delta_p u = u^{p^*-1} h(u), \qquad 1<p<n, $$ where $h$ is a positive, bounded, continuous, and nonincreasing function. Our first main result is a complete classification of normalized, bounded, positive entire solutions for every equation...
We study global regularity of solutions to Dirichlet or Neumann elliptic problems in spherical sectors $S_{D,R}$ of radius $R>0$ in $\mathbb{R}^N, N\ge 2$, where $D$ is the bounded domain on the unit sphere $\mathbb{S}^{N-1}$ which spans the spherical sector. One of the main results shows that boundedness of the gradie...
Carlo Alberto Antonini, F. Pacella, Camilla Chiara Polvara et al.· 0 citations
For every $n\geq 3$, we construct a nonconstant real-valued function $U\in C^\infty(\overline{\mathbb R^n_+})$, harmonic in the upper half-space, whose complete boundary jet vanishes on a compact nowhere dense subset of $\partial\mathbb R^n_+$ of positive $(n-1)$-dimensional measure. The essential construction takes pl...
Guo-Lin Qin, Wei-Cheng Zhan· 0 citations
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