Related papers: A non-local inequality and global existence
We study the existence and multiplicity results for the following nonlocal $p(x)$-Kirchhoff problem: \begin{equation} \label{10} \begin{cases} -\left(a-b\int_\Omega\frac{1}{p(x)}| \nabla u| ^{p(x)}dx\right)div(|\nabla u| ^{p(x)-2}\nabla…
This paper is devoted to the theoretical analysis of the nonlinear plate equations in $\mathbb{R}^{n}\times (0,\infty),$ $n\geq1,$ with nonlinearity involving a type polynomial behavior. We prove the existence and uniqueness of global mild…
The Cauchy problem in $\mathbb{R}^d,$ $d\geq 1,$ for a non-local in time p-Laplacian equations is considered. The nonexistence of nontrivial global weak solutions by using the test function method is obtained.
We obtain a generalization of the Picone inequality which, in combination with the classical Picone inequality, appears to be useful for problems with the $(p,q)$-Laplace type operators. With its help, as well as with the help of several…
We study the existence of nontrivial nonlocal nonnegative solutions $u(x,t)$ of the nonlinear initial value problems \[ (\partial_t -\Delta)^\alpha u\geq u^\lambda \quad \text{in } \mathbb{R}^n \times\mathbb{R},\,n\geq 1 \] \[ u=0…
We establish both the existence and uniqueness of non-negative global solutions for the nonlinear heat equation $u_t-\Delta u=|x|^{-\gamma}\,u^q$, $0<q<1$, $\gamma>0$ in the whole space $\mathbb{R}^N$, and for non-negative initial data…
By topological arguments, we prove new results on the existence, non-existence, localization and multiplicity of nontrivial solutions of a class of perturbed nonlinear integral equations. These type of integral equations arise, for example,…
We study sharp conditions for the existence and nonexistence of infinitely many nonnegative solutions to the problem $-\Delta_p u = \lambda f(u)$ in a bounded domain with Dirichlet boundary conditions, where $f$ is a continuous function…
We prove an energy inequality for nonlocal diffusion operators of the following type, and some of its generalisations: $Lu (x) := \int_{\mathbb{R}^N} K(x,y) (u(y) - u(x)) dy$, where $L$ acts on a real function $u$ defined on $\mathbb{R}^N$,…
We study the existence of global-in-time solutions for a nonlinear heat equation with nonlocal diffusion, power nonlinearity and suitably small data (either compared pointwisely to the singular solution or in the norm of a critical Morrey…
In this work we study the global existence for 3d semilinear wave equation with non-negative potential satisfying generic decay assumptions. In the supercritical case we establish the small data global existence result. The approach is…
\begin{abstract} In this paper, we consider the following system of difference equations \begin{equation*} x_{n+1}=\alpha+\dfrac{y_{n}^p}{y_{n-2}^p},\ y_{n+1}=\alpha+ \dfrac{x_{n}^q}{x_{n-2}^q}, \ n=0, 1, 2, ... \end{equation*} where…
In this work, we consider a mixed local and nonlocal Dirichlet problem with supercritical nonlinearity. We first establish a multiplicity result for the problem \begin{equation} Lu=|u|^{p-2}u+\mu|u|^{q-2}u~~\text{in}~~\Omega,~~~~~…
We study the limit, when $k\to\infty$ of solutions of $u_t-\Delta u+f(u)=0$ in $R^N\times(0,\infty)$ with initial data $k\gd$, when $f$ is a positive increasing function. We prove that there exist essentially three types of possible…
We are mainly concerned with the nonlinear $p$-Laplace equation \begin{equation*} -\Delta_pu+\rho|u|^{p-2}u=\psi(x,u) \end{equation*} on a locally finite graph $G=(V,E)$, where $p$ belongs to $(1, +\infty)$. We obtain existence of positive…
We study a singular elliptic problem driven by a mixed local-nonlocal operator of the form \begin{equation*} \begin{aligned} -\Delta_p u + (-\Delta_q)^s u &= \frac{\lambda}{u^{\delta}} + u^r \text{ in } \Omega\newline u > 0 \text{ in }…
In this paper, using the theory developed in [8], we obtain some results of a totally new type about a class of non-local problems. Here is a sample: Let $\Omega\subset {\bf R}^n$ be a smooth bounded domain, with $n\geq 4$, let $a, b,…
In this article, we show the global multiplicity result for the following nonlocal singular problem \begin{equation*} (P_\la):\;\quad (-\De)^s u = u^{-q} + \la u^{{2^*_s}-1}, \quad u>0 \; \text{in}\; \Om,\quad u = 0 \; \mbox{in}\; \mb R^n…
We study regularity properties for solutions to the nakedly degenerate elliptic equation $a_{ij}\partial_{ij}u =0$, where the coefficients satisfy $I \ge a_{ij}(x) \ge \lambda(x) I$ and the only assumption is that $\lambda^{-1} \in L^p$. We…
We establish non-existence results for the Cauchy problem of some semilinear heat equations with non-negative initial data and locally Lipschitz, nonnegative source term $f$. Global (in time) solutions of the scalar ODE $\dot v=f(v)$ exist…