Related papers: Lyapunov inequalities for Partial Differential Equ…

This note introduces a class of nonlinear Neumann problems on balls expanding with the radii tending towards infinity. Performing singular perturbation arguments, we establish the corresponding concentration phenomenon and refined…

In this short note, we present a Lyapunov-type inequality that corrects the recently obtained result in [M. Kirane, B. T. Torebek: A Lyapunov-type inequality for a fractional boundary value problem with Caputo-Fabrizio derivative, J. Math.…

The first nontrivial eigenfunction of the Neumann eigenvalue problem for the $p$-Laplacian, suitable normalized, converges as $p$ goes to $\infty$ to a viscosity solution of an eigenvalue problem for the $\infty$-Laplacian. We show among…

We find a criterion for correct solvability in L_p(R) of a linear differential equation of a first order with non-negative locally integrated coefficient and study the asymptotic properties of its solutions.

We study the existence of Fuchsian differential equations in positive characteristic with nilpotent p-curvature, and given local invariants. In the case of differential equations with logarithmic local mononodromy, we determine the minimal…

The continuous-time differential Lyapunov equation is widely used in linear optimal control theory, a branch of mathematics and engineering. In quantum physics, it is known to appear in Markovian descriptions of linear (quadratic…

We are interested in the structure of the positive radial solutions of the supercritical Neumann problem $\varepsilon^2\Delta u-u+u^p=0$ on a unit ball in $\mathbb{R}^N$ , where $N$ is the spatial dimension and $p>p_S:=(N+2)/(N-2)$, $N\ge…

Interpolation inequalities for $C^m$ functions allow to bound derivatives of intermediate order $0 < j<m$ by bounds for the derivatives of order $0$ and $m$. We review various interpolation inequalities for $L^p$-norms ($1 \le p \le…

In this work, we study the existence of weak solution to the following quasi linear elliptic problem involving the fractional $p$-Laplacian operator, a Hardy potential and multiple critical Sobolev nonlinearities with singularities,…

In this paper, we establish gradient estimates for positive solutions to the following equation with respect to the $p$-Laplacian $$\Delta_{p}u=-\lambda |u|^{p-2}u$$ with $p>1$ on a given complete Riemannian manifold. Consequently, we…

The class of problems treated here are elliptic partial differential equations with a homogeneous boundary condition and a non-linear perturbation obtained by composition with a fixed smooth function. The existence of solutions is obtained…

We study a non-local eigenvalue problem related to the fractional Sobolev spaces for large values of p and derive the limit equation as p goes to infinity. Its viscosity solutions have many interesting properties and the eigenvalues exhibit…

In the past years, the phenomenon of fractional regularity has been addressed for a large class of linear and/or quasilinear differential operators, mostly, in terms of certain Besov spaces. As it turned out, for equations governed by the…

In this paper, we investigate critical quasilinear elliptic partial differential equations on a complete Riemannian manifold with nonnegative Ricci curvature. By exploiting a new and sharp nonlinear Kato inequality and establishing some…

If $(M,g)$ is a compact Riemannian manifold of dimension $n\ge 2$ we give necessary and sufficient conditions for improved $L^p(M)$-norms of eigenfunctions for all $2<p\ne p_c=\tfrac{2(n+1)}{n-1}$, the critical exponent. Since improved…

The present paper studies the fractional $p$-Laplacian boundary value problems with jumping nonlinearities at zero or infinity and obtain the existence of multiple solutions and sign-changing solutions by constructing the suitable…

We are concerned with the solvability of linear second order elliptic partial differential equations with nonlinear boundary conditions at resonance, in which the nonlinear boundary conditions perturbation is not necessarily required to…

While it is known that one can consider the Cauchy problem for evolution equations with Caputo derivatives, the situation for the initial value problems for the Riemann-Liouville derivatives is less understood. In this paper we propose new…

This paper is devoted to studying the local behavior of non-negative weak solutions to the doubly non-linear parabolic equation \begin{equation*} \partial_t u^q - \text{div}\big(|D u|^{p-2}D u\big) = 0 \end{equation*} in a space-time…

We derive a priori bounds for positive supersolutions of $ - \Delta_{p} u = \rho(x) f(u) $, where $p>1$ and $\Delta_{p}$ is the $p$-Laplace operator, in a smooth bounded domain of $R^{N}$ with zero Dirichlet boundary conditions. We apply…