Related papers: Singular solutions to the heat equations with nonl…

We obtain necessary and sufficient conditions for the existence of a positive finite energy solution to the inhomogeneous quasilinear elliptic equation \[ -\Delta_{p} u = \sigma u^{q} + \mu \quad \text{on} \;\; \mathbb{R}^n \] in the…

In this paper, we consider the following semi-linear complex heat equation \begin{eqnarray*} \partial_t u = \Delta u + u^p, u \in \mathbb{C} \end{eqnarray*} in $\mathbb{R}^n,$ with an arbitrary power $p,$ $ p > 1$. In particular, $p$ can be…

We study the existence, multiplicity, and certain qualitative properties of solutions to the zero Dirichlet problem for the equation $-\Delta_p u = \lambda |u|^{p-2}u + a(x)|u|^{q-2}u$ in a bounded domain $\Omega \subset \mathbb{R}^N$,…

In this paper we show that the number of radial positive solutions of the following critical problem $$ \Delta_p u(x) + \lambda K(|x|) \,u(x) \, |u(x)|^{q-2} =0\,,$$ $$ u(x)>0 \quad |x|<1,$$ $$ u(x)=0 \quad |x|=1,$$ where $q=…

In this paper, we are devoted to studying the positive solutions of the following higher order Hardy-H\'enon equation $$ (-\Delta)^{m}u=|x|^{\alpha}u^{p} \quad\mbox{in}~ B_{1}\setminus\{0\}\subset\mathbb{R}^{n} $$ with an isolated…

In this paper we analyze the asymptotic behaviour as $p\to 1^+$ of solutions $u_p$ to $$ \left\{ \begin{array}{rclr} -\Delta_pu&=&\lambda|\nabla u|^{p-2}\nabla u\cdot\frac{x}{|x|^2}+ f&\quad \mbox{ in } \Omega,\\ u_p&=&0 &\quad \mbox{ on…

We consider the problem of existence of a solution $u$ to $\partial_t u-\partial_{xx} u = 0$ in $(0,T)\times\mathbb{R}_+$ subject to the boundary condition $-u_x(t,0)+g(u(t,0))=\mu$ on $(0,T)$ where $\mu$ is a measure on $(0,T)$ and $g$ a…

In this paper we consider nonlinear elliptic PDEs of the type $$-\Delta_p u+a(x)|u|^{p-2}u=|u|^{p^*-2}u \qquad \mbox{ in }\Omega,$$ where $1<p<N$ and $p^*=Np/(N-p)$ is the critical Sobolev exponent, and allowing the asymptotic behavior of…

Let $p\in(0,\frac{N}{N-2\alpha})$, $\alpha\in(0,1)$ and $\Omega\subset \R^N$ be a bounded $C^2$ domain containing $0$. If $\delta_0$ is the Dirac measure at $0$ and $k>0$, we prove that the weakly singular solution $u_k$ of $(E_k)$ $…

We study the existence/nonexistence and qualitative properties of the positive solutions to the problem \begin{align*} (-\Delta)^s u -\theta\frac{u}{|x|^{2s}}&=u^p - u^q \quad\text{in }\,\, \mathbb{R}^N,\quad u > 0 \quad\text{in }\,\,…

We study the singularity formation of strong solutions to the two-dimensional (2D) Cauchy problem of the non-baratropic compressible magnetohydrodynamic equations without heat conductivity. It is proved that the strong solution exists…

In this paper we shall investigate the nonexistence of positive solutions for the following nonlinear parabolic partial differential equation:\[ \begin{cases} \frac{\partial u}{\partial t}= \Delta_{\mathbb{G},p}u+V(x)u^{p-1} &…

In this article, we study the existence and multiplicity of solutions of the following $(p,q)$-Laplace equation with singular nonlinearity: \begin{equation*} \left\{\begin{array}{rllll} -\Delta_{p}u-\ba\Delta_{q}u & = \la u^{-\de}+ u^{r-1},…

In this paper we consider the class $\mathcal{A}$ of those solutions $u(x,t)$ to the conjugate heat equation $\frac{d}{dt}u = -\Delta u + Ru$ on compact K\"ahler manifolds $M$ with $c_1 > 0$ (where $g(t)$ changes by the unnormalized…

We study the existence and nonexistence of positive (super-)solutions to a singular semilinear elliptic equation $$-\nabla\cdot(|x|^A\nabla u)-B|x|^{A-2}u=C|x|^{A-\sigma}u^p$$ in cone--like domains of $\R^N$ ($N\ge 2$), for the full range…

In this paper we consider a superlinear one-dimensional elliptic boundary value problem that generalizes the one studied by Moore and Nehari in [43]. Specifically, we deal with piecewise-constant weight functions in front of the…

In this paper, we classify the singularities of nonnegative solutions to fractional elliptic equation \begin{equation}\label{eq 0.1} \arraycolsep=1pt \begin{array}{lll} \displaystyle (-\Delta)^\alpha u=u^p\quad &{\rm in}\quad…

We study the existence of nonnegative solutions to the following nonlocal elliptic problem involving singularity \begin{align} \mathfrak{M}\left(\int_{Q}\frac{|u(x)-u(y)|^p}{|x-y|^{N+ps}}dxdy\right)(-\Delta)_{p}^{s}…

We study the Dirichlet problem of the following discrete infinity Laplace equation on unbounded subgraphs \begin{equation*} \Delta_{\infty}u(x):=\inf_{y\sim x}u(y)+\sup_{y\sim x}u(y)-2u(x)=f(x). \end{equation*} For the homogeneous case…

We generalise and sharpen several recent results in the literature regarding the existence and complete classification of the isolated singularities for a broad class of nonlinear elliptic equations of the form \begin{equation} -{\rm…