相关论文: Blowing-up Solutions with Residual Mass in a Sligh…

We consider the semilinear heat equation $u_t=\Delta u+|u|^{p-1} u$ in possibly non-convex and unbounded domains. Our main result shows the nonexistence of type II blow-up for possibly sign-changing solutions in the energy subcritical range…

In this paper, we investigate the initial boundary value problem of the following nonlinear extensible beam equation with nonlinear damping term $$u_{t t}+\Delta^2 u-M\left(\|\nabla u\|^2\right) \Delta u-\Delta u_t+\left|u_t\right|^{r-1}…

We consider radial solutions of the slightly subcritical problem $-\Delta u_\varepsilon = |u_\varepsilon|^{\frac{4}{n-2}-\varepsilon}u_\varepsilon$ either on $\mathbb R^n$ ($n\geq 3$) or in a ball $B$ satisfying Dirichlet or Neumann…

We investigate in this work families $(u_\epsilon)_{\epsilon >0}$ of sign-changing blowing-up solutions of asymptotically critical stationary nonlinear Schr\"odinger equations of the following type: $$\Delta_g u_\epsilon + h_\epsilon…

The parabolic problem $u_t-\Delta u=\frac{\lambda f(x)}{(1-u)^2}+P$ on a bounded domain $\Omega$ of $R^n$ with Dirichlet boundary condition models the microelectromechanical systems(MEMS) device with an external pressure term. In this…

In the present paper, we study the existence and uniqueness of solutions to some nonlocal singular elliptic problem under Dirichlet boundary condition. Problem is settled in Musielak-Sobolev spaces.

Let $(M,g)$ be a $n-$dimensional compact Riemannian manifold with boundary. We consider the Yamabe type problem \begin{equation} \left\{ \begin{array}{ll} -\Delta_{g}u+au=0 & \text{ on }M \\ \partial_\nu u+\frac{n-2}{2}bu= u^{{n\over…

The initial boundary-value problem (IBVP) and the Cauchy problem for the Kuramoto--Sivashinsky equation and other related $2m$th-order semilinear parabolic partial differential equations in one and N dimensions are considered. Global…

In this paper we perform a fine blow-up analysis for a fourth order elliptic equation involving critical Sobolev exponent, related to the prescription of some conformal invariant on the standard sphere. We derive from this analysis some a…

We consider the nonlinear eigenvalue problem $ L u = \lambda f(u) $, posed in a smooth bounded domain $ \Omega \subseteq \Bbb{R}^{N} $ with Dirichlet boundary condition, where $ L $ is a uniformly elliptic second-order linear differential…

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…

We consider the $L^2$-critical nonlinear Schr\"odinger equation (NLS) with the delta potential $$i\partial_tu +\partial^2_x u + \mu \delta u +|u|^{4}u=0, \, \, t\in \R, \, x\in \R , $$ where $ \mu \in \R$, and $\delta$ is the Dirac delta…

We consider the nonlinear Schr\"odinger equation $iu_t=-\Delta u-|u|^{p-1}u$ in dimension $N\geq 3$ in the $L^2$ super critical range $1+\frac{4}{N}<p<\frac{N+2}{N-2}$. The corresponding scaling invariant space is $\dot{H}^{s_c}$ with…

We investigate bubbling solutions for the nonlocal equation \[ A_\Omega^s u =u^p,\ u >0 \quad \mbox{in } \Omega, \] under homogeneous Dirichlet conditions, where $\Omega$ is a bounded and smooth domain. The operator $A_\Omega^s$ stands for…

We investigate the weak solvability and properties of weak solutions to the Dirichlet problem for a scalar elliptic equation $-\Delta u + b^{(\alpha)}\cdot \nabla u= f$ in a bounded domain $\Omega\subset {\mathbb R^2}$ containing the…

In this paper we are interested in positive classical solutions of \begin{equation} \label{eqx} \left\{\begin{array}{ll} -\Delta u = a(x) u^{p-1} & \mbox{ in } \Omega, \\ u>0 & \mbox{ in } \Omega, \\ u= 0 & \mbox{ on } \pOm, \end…

Our main result shows that the mass $2\pi$ is critical for the minimal Keller-Segel system \begin{align}\label{prob:abstract}\tag{$\star$} \begin{cases} u_t = \Delta u - \nabla \cdot (u \nabla v), \\ v_t = \Delta v - v + u, \end{cases}…

The parabolic-elliptic cross-diffusion system \[ \left\{ \begin{array}{l} u_t = \Delta u - \nabla \cdot \Big(uf(|\nabla v|^2) \nabla v \Big), \\[1mm] 0 = \Delta v - \mu + u, \qquad \int_\Omega v=0, \qquad \mu:=\frac{1}{|\Omega|} \int_\Omega…

For an integer $n \ge 3$ and any positive number $\epsilon$ we establish the existence of smooth functions K on $R^n \setminus \{0 \}$ with $|K - 1| \le \epsilon$, such that the equation $\Delta u + n (n - 2) K u^{{n + 2}\over {n - 2}} = 0$…

The present paper is concerned with the half-space Dirichlet problem \begin{equation} \tag{$P_c$} \label{problem-abstract} -\Delta v + v = |v|^{p-1}v,\ \mbox{ in } \mathbb{R}^N_{+}, \qquad v = c,\ \mbox{ on } \partial \mathbb{R}^N_{+},\…