Related papers: The Singular Set of Higher Dimensional Unstable Ob…

Our purpose of this paper is to study isolated singular solutions of semilinear Helmholtz equation $$ -\Delta u-u=Q|u|^{p-1}u \quad{\rm in}\ \ \mathbb{R}^N\setminus\{0\},\ \qquad\lim_{|x|\to0}u(x)=+\infty, $$ where $N\geq 2$, $p>1$ and the…

In this article we consider the existence of positive singular solutions on bounded domains and also classical solutions on exterior domains. First we consider positive singular solutions of the following problems: \begin{equation}…

We consider a parabolic equation of the form u_t=\Delta u +f(u)+h(x,t) in R^N\times (0,\infty), where f in C^1(R) is such that f(0)=0 and f'(0)<0 and h is a suitable function on R^N\times (0,\infty). We show that under certain conditions,…

We establish that the initial value problem for the quadratic non-linear Schr\"odinger equation $$ iu_t - \Delta u = u^2$$ where $u: \R^2 \times \R \to \C$, is locally well-posed in $H^s(\R^2)$ when $s > -1$. The critical exponent for this…

We consider the diffusive Hamilton-Jacobi equation $u_t - \Delta u = |\nabla u|^p$ in a bounded planar domain with zero Dirichlet boundary condition. It is known that, for $p>2$, the solutions to this problem can exhibit gradient blow-up…

We study the nodal set of solutions to equations of the form $$ (-\Delta)^s u = \lambda_+ (u_+)^{q-1} - \lambda_- (u_-)^{q-1}\quad\text{in $B_1$}, $$ where $\lambda_+,\lambda_->0, q \in [1,2)$, and $u_+$ and $u_-$ are respectively the…

In this paper we consider the higher order Lioville-type equation $(-\Delta)^{m} u=\rho^{2m} V(x) e^{u}$ in $\Omega\subseteq\mathbb{R}^{2m}$ with $V\neq0$ a given smooth potential, $\rho\in\mathbb{R}^{+}$ a small parameter which tends to…

We give a new bound on the exponent for the nonexistence of stable solutions to the biharmonic problem $$\Delta^2u=u^p,\quad u>0 in \mathbb{R}^n $$ where $p>1, n \geq 20$.

We study the behavior of weak solutions to the singular quasilinear elliptic problem $-\Delta_p u + \vartheta |\nabla u|^q = \frac{1}{u^\gamma} + f(u)$, in a bounded domain with the Dirichlet boundary condition, where $p>1$, $\gamma>0$,…

We study the existence and uniqueness of new classes of solutions of the superlinear equation $-\Delta u+u^q=0$ (q>1) in a domain of R^N or in a finely open set for the topology associated to the Bessel capacity C_{2,q'}. Condition of…

In this paper, we prove the existence and uniqueness of $W^{2,p}$ ($n<p<\infty$) solutions of a double obstacle problem with $C^{1,1}$ obstacle functions. Moreover, we show the optimal regularity of the solution and the local $C^1$…

We consider the problem -\Delta u - g(u) = \lambda u, u \in H^1(\R^N), \int_{\R^N} u^2 = 1, \lambda\in\R, in dimension $N\ge2$. Here $g$ is a superlinear, subcritical, possibly nonhomogeneous, odd nonlinearity. We deal with the case where…

This paper investigates the existence of positive solutions for regular discrete second-order single-variable boundary value problems with mixed boundary conditions, including a nonhomogeneous Dirichlet boundary condition, of the form:…

In this paper we are concerned with a general singular Dirichlet boundary value problem whose model is the following $$ \begin{cases} -\Delta u = \frac{\mu}{u^{\gamma}} & \text{in}\ \Omega, u=0 &\text{on}\ \partial\Omega, u>0 &\text{on}\…

This paper is concerned with the Dirichlet problem for an equation involving the 1--Laplacian operator $\Delta_1 u$ and having a singular term of the type $\frac{f(x)}{u^\gamma}$. Here $f\in L^N(\Omega)$ is nonnegative, $0<\gamma\le1$ and…

We study the regularity of the extremal solution of the semilinear biharmonic equation $\bi u=\f{\lambda}{(1-u)^2}$, which models a simple Micro-Electromechanical System (MEMS) device on a ball $B\subset\IR^N$, under Dirichlet boundary…

We analyze stable weak solutions to the fractional Gel'fand problem \begin{equation*} (-\Delta)^su=e^u\quad\mathrm{in}\quad \Omega\subset\mathbb{R}^n. \end{equation*} We prove that the dimension of the singular set is at most $n-10s.$

Let $u$ be a harmonic function in a $C^1$-Dini domain $D$ such that $u$ vanishes on a boundary surface ball $\partial D \cap B_{5R}(0)$. We consider an effective version of its singular set (up to boundary) $\mathcal{S}(u):=\{X\in…

We show that singular sets of free boundaries arising in codimension one anisotropic geometric variational problems are $\mathcal H ^{n-3}$-negligible, where $n$ is the ambient space dimension. In particular our results apply to capillarity…

We consider the Cauchy-problem for the following parabolic equation: \begin{equation*} \displaystyle u_t = \Delta u+ f(u,|x|), \end{equation*} where $x \in \mathbb{R}^n$, $n >2$, and $f=f(u,|x|)$ is either critical or supercritical with…