Related papers: Maximal solutions of equation u = uq in arbitrary …

The purpose of this paper is to give a necessary and sufficient condition for the existence and non-existence of global solutions of the following semilinear parabolic equations \[ u_{t}=\Delta u+\psi(t)f(u),\,\,\mbox{ in }\Omega\times…

Let $\alpha$ be an algebraic number of degree $d\ge 3$ having at most one real conjugate and let $K$ be the algebraic number field ${\mathbf Q}(\alpha)$. For any unit $\epsilon$ of $K$ such that ${\mathbf Q}(\alpha\epsilon)=K$, we consider…

In this paper, the fully parabolic Keller-Segel system \begin{equation} \left\{ \begin{array}{llc} u_t=\Delta u-\nabla\cdot(u\nabla v), &(x,t)\in \Omega\times (0,T),\\ v_t=\Delta v-v+u, &(x,t)\in\Omega\times (0,T),\\ \end{array} \right.…

In this paper, the fractional order curvature equation $(-\Delta)^\gamma u = (1 + \varepsilon K(x))u^{\frac{N + 2\gamma}{N - 2\gamma}}$ in $\mathbb{R}^N$ is considered. Assuming $K(x)$ has two critical points satisfying certain local…

We study metrics of constant $Q$-curvature in the Euclidean space with a prescribed singularity at the origin, namely solutions to the equation $$(-\Delta)^\frac{n}{2}w=e^{nw}-c\delta_{0} \text{ on } \mathbb R^n,$$ under a finite volume…

We consider the cubic and quintic nonlinear Schr\"{o}dinger equations (NLS) under the $\mathbb{R}^{d}$ and $\mathbb{T}^{d}$ energy-supercritical setting. Via a newly developed unified scheme, we prove the unconditional uniqueness for…

In this paper, we are concerned with the following elliptic equation \begin{equation*} \begin{cases} -\Delta u= Q(x)u^{2^*-1 }+\varepsilon u^{s},~ &{\text{in}~\Omega},\\[1mm] u>0,~ &{\text{in}~\Omega},\\[1mm] u=0, &{\text{on}~\partial…

We establish nonuniqueness of solutions for Cauchy problems of semilinear heat equations with a wide class of nonlinearities. Specifically, we consider \[ \begin{cases} \partial_tu-\Delta u=f(u), & x\in\mathbb{R}^N,\ t>0,\\ u(x,0)=u_0(x), &…

We consider the following prescribed boundary mean curvature problem in $\mathbb B^N$ with the Euclidean metric $-\Delta u =0$, $u>0$ in $B^N, \frac{\partial u}{\partial\nu} + \frac{N-2}{2} u =\frac{N-2}{2} K(x) u^{N/(N-2)}$ on $S^{N-1},…

We prove comparison, uniqueness and existence results for viscosity solutions to a wide class of fully nonlinear second order partial differential equations $F(x, u, du, d^{2}u)=0$ defined on a finite-dimensional Riemannian manifold $M$.…

We consider quasilinear elliptic problems of the form \[ -\operatorname{div}\big(\phi(|\nabla u|)\nabla u\big)+V(x)\phi (|u|)u=f(u)\qquad u\in W^{1,\Phi}(\mathbb{R}^{N}), \] where $\phi$ and $f$ satisfy suitable conditions. The positive…

The singular parabolic problem $u_t=\Delta u -\frac{\lambda f(x)}{(1+u)^2}$ on a bounded domain $\Omega$ of $R^N$ with Dirichlet boundary conditions, models the dynamic deflection of an elastic membrane in a simple electrostatic…

In this paper, we investigate the quantitative unique continuation, propagation of smallness and measure bounds of nodal sets of solutions to the Buckling type equation $\triangle^2u+\lambda\triangle u-k^2u=0$ in a bounded analytic domain…

In this paper, we consider the following Klein-Gordon-Maxwell equations \begin{eqnarray*} \left\{ \begin{array}{ll} -\Delta u+ V(x)u-(2\omega+\phi)\phi u=f(x,u)+h(x)&\mbox{in $\mathbb{R}^{3}$},\\ -\Delta \phi+ \phi u^2=-\omega u^2&\mbox{in…

We consider the problem $(P)$, $$ -\Delta u =c(x)u+\mu|\nabla u|^2 +f(x), \quad u \in H^1_0(\Omega) \cap L^{\infty}(\Omega),$$ where $\Omega$ is a bounded domain of $\mathbb{R}^N$, $N \geq 3$, $\mu>0, \, c \in…

We study quasilinear elliptic equations of the type $-\Delta_{p} u = \sigma u^{q} + \mu \; \; \text{in} \;\; \bf{R}^n$ in the case $0<q< p-1$, where $\mu$ and $\sigma$ are nonnegative measurable functions, or locally finite measures, and…

We construct functions $u: \mathbb{R}^2 \to \mathbb{C}$ that satisfy an elliptic eigenvalue equation of the form $-\Delta u + W \cdot \nabla u + V u = \lambda u$, where $\lambda \in \mathbb{C}$, and $V$ and $W$ satisfy $|V(x)| \lesssim…

We investigate a class of Kirchhoff type equations involving a combination of linear and superlinear terms as follows: \begin{equation*} -\left( a\int_{\mathbb{R}^{N}}|\nabla u|^{2}dx+1\right) \Delta u+\mu V(x)u=\lambda…

We study the limit, when $k\to\infty$, of the solutions $u=u_{k}$ of (E) $\prt_{t}u-\Delta u+ h(t)u^q=0$ in $\BBR^N\ti (0,\infty)$, $u_{k}(.,0)=k\delta_{0}$, with $q>1$, $h(t)>0$. If $h(t)=e^{-\gw(t)/t}$ where $\gw>0$ satisfies to…

In this paper, we consider the $k$-Hessian equation $S_{k}(D^{2}u)=b(x)f(u)\mbox{ in }\Omega,\,u=+\infty \mbox{ on }\partial\Omega$, where $\Omega$ is a smooth, bounded, strictly convex domain in $\mathbb{R}^{N}$ with $N\geq2$, $b\in \rm…