In this paper, we study local and global properties of positive solutions to the equation $-\Delta u=u^p|\nabla u|^q$ in a domain $\Omega$ of $\mathbb R^N$, where $p$ and $q$ are parameters. We introduce a linear operator to construct the differential inequality to obtain gradient estimates, and further establish Liouville-type theorems. As an application, we derive universal estimates for local solutions. Some of our results are new, as we extend the condition $p+q<(N+3)/(N-1)$ considered by He, Hu and Wang [Math. Z. 313 (2026), No. 6] to a wider range of parameters.
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...
In this paper, we investigate the semilinear parabolic equation $\partial_t f-\Delta f=a(x,t)f^p$ on a complete Riemannian manifold with Ricci curvature bounded from below. By means of Nash-Moser iteration, we establish Li-Yau type gradient estimates for positive solutions to this equation, where the coefficient functi...
We study positive entire solutions of the flat constant $Q/R$-curvature equation in dimension $n\geq5$. We prove that the weak superharmonicity condition $-\Delta u\geq0$ already forces every nonconstant solution to be admissible. With this automatic admissibility, we classify all such solutions as positive constants o...
We prove a sharp nonlinear version of the celebrated Li-Yau inequality for positive solutions of the normalized parabolic $p$-Laplacian equation $u_t = \Delta u + (p-2)|\nabla u|^{-2}\nabla^2u(\nabla u,\nabla u)$, $1<p<\infty$, on a closed Riemannian manifold with nonnegative Ricci curvature, the equation being interpr...
We study positive supersolutions of quasilinear elliptic problems driven by the (p, q)-Laplace operator which arise in models with nonstandard and unbalanced growth. Our approach is based on an explicit construction of radial solutions and torsion functions that serve as comparison functions. This gives a fundamental p...
A. Aghajani, C. Cowan, J. Kinnunen· Manuscripta mathematica· 0 citations
We establish an intrinsic Harnack inequality with an optimal tail term for weak solutions of parabolic fractional $p$-Laplace equations with $p>2$ and $s\in(0,1)$ whose kernels are symmetric, measurable, and comparable to $|x-y|^{-n-sp}$. It constitutes the nonlocal analogue of the intrinsic Harnack inequality of DiBen...
Harsh Prasad· 0 citations
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