We investigate interior regularity for stable solutions of the semilinear elliptic equation $-\Delta u=f(u)$ in an open set $\Omega\subseteq \mathbb R^n$, where $f$ is locally Lipschitz and may change sign. We establish sharp interior H\"older estimates for stable classical solutions in every dimension $n\le 9$, thereby fully resolving a question raised by Cabr\'e concerning interior regularity without the nonnegativity assumption on $f$. For $n\le 5$, we also give a short alternative proof of the corresponding H\"older regularity result due to Peng, Zhang, and Zhou. We further establish higher-integrability estimates for the gradient and $L^1$ bounds for the Hessian in all dimensions. For $n\ge 10$, we obtain BMO and Morrey estimates, extending related results previously established for nonnegative nonlinearities. When $f$ is convex, we extend the interior regularity estimates to stable weak solutions through an approximation theorem. As applications, we prove Liouville-type theorems for entire stable solutions of equations with sign-changing nonlinearities across all dimensions.
We establish universal interior H\"older and fractional energy estimates for stable solutions of $(-\Delta)^su=f(u)$, $0<s<1$, with $f\in C^1_{\mathrm{loc}}(\mathbb{R})$ nonnegative and convex. The bounds depend only on a weighted $L^1$ norm of the solution and hold under optimal conditions on the dimension and the fra...
We study local and global properties of positive solutions to the equation $-\Delta u=u^p+M|\nabla u|^q$ in a domain $\Omega$ of $\mathbb R^N$, where $p,q$ are parameters and $M>0$. By constructing a linear operator, we establish the differential inequality containing an auxiliary function. By selecting appropriate aux...
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...
We establish an interior $C^2$ estimate for locally bounded plurisubharmonic Bedford-Taylor solutions of the complex Monge--Amp\`ere equation $\det(u_{i\bar j})=f$ under weak quadratic growth bounds on sufficiently large balls, where $f$ is positive and Dini-continuous. We assume a distributional lower bound for the La...
We investigate the regularity of viscosity solutions to a class of nonlocal Hamilton-Jacobi equations driven by x-dependent integro-differential operators and coercive superlinear Hamiltonians. We first establish H{\"o}lder regularity for bounded viscosity solutions under general structural and continuity assumptions o...
A. Ciomaga, T. M. Lê, Olivier Ley et al.· 0 citations
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...
Chong Gu· 0 citations
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