Related papers: Non-Linear Effects in a Yamabe-Type Problem with Q…

In this paper we study the free boundary regularity for almost-minimizers of the functional \begin{equation*} J(u)=\int_{\mathcal O} |\nabla u(x)|^2 +q^2_+(x)\chi_{\{u>0\}}(x) +q^2_-(x)\chi_{\{u<0\}}(x)\ dx \end{equation*} where $q_\pm \in…

We characterize the existence of solutions to the quasilinear Riccati type equation \begin{eqnarray*} \left\{ \begin{array}{rcl} -{\rm div}\,\mathcal{A}(x, \nabla u)&=& |\nabla u|^q + \sigma \quad \text{in} ~\Omega, \\ u&=&0 \quad…

In this paper, we prove the existence of weak solutions for the following nonlinear elliptic system {lll} -\Delta_{p(x)}u = a(x)|u|^{p(x)-2}u - b(x)|u|^{\alpha(x)}|v|^{\beta(x)} v + f(x) in \Omega, \Delta_{q(x)}v = c(x) |v|^{q(x)-2}v -…

Let $\Omega\subset\mathbb{R}^N$, $N\geq 2$, be an open bounded connected set. We consider the fractional weighted eigenvalue problem $(-\Delta)^s u =\lambda \rho u$ in $\Omega$ with homogeneous Dirichlet boundary condition, where…

We consider elliptic equations with non-Lipschitz nonlinearity $$ -\Delta u = \lambda |u|^{\beta-1}u-|u|^{\alpha-1}u$$ in a smooth bounded domain $\Omega \subset \mathbb{R}^n$, $n\geq 3$, with Dirichlet boundary conditions; here…

We consider the minimizers of the energy $$ \|u\|_{H^s(\Omega)}^2+\int_\Omega W(u)\,dx,$$ with $s \in (0,1/2)$, where $\|u\|_{H^s(\Omega)}$ denotes the total contribution from $\Omega$ in the $H^s$ norm of $u$, and $W$ is a double-well…

We consider the following Lane-Emden system with Neumann boundary conditions \[ -\Delta u= |v|^{q-1}v \text{ in } \Omega,\qquad -\Delta v= |u|^{p-1}u \text{ in } \Omega,\qquad \partial_\nu u=\partial_\nu v=0 \text{ on } \partial \Omega, \]…

In this paper, we study a solvability result for the nonlinear problem $$ \mbox {div } \left ( \vert \nabla_\omega u\vert^{p-2}\nabla_\omega u \right )+v(x) u^{q-1}+\mu u^{\gamma-1}=0, \quad z\in \Omega, \quad u \Big \vert_{\partial…

Let $N>2$, $p\in \left(\frac{2N}{N+2},+\infty\right)$, and $\Omega$ be an open bounded domain in $\mathbb{R}^N$. We consider the minimum problem $$ \mathcal{J} (u) := \displaystyle\int_{\Omega } \left(\frac{1}{p}| \nabla u|…

We study a conormal boundary value problem for a class of quasilinear elliptic equations in bounded domain $\Omega$ whose coefficients can be degenerate or singular of the type $\text{dist}(x, \partial \Omega)^\alpha$, where $\partial…

Consider the quasilinear diffusion problem \[\begin{cases}\mathbf{u}'+\Pi(t,x,\mathbf{u},\Sigma \mathbf{u})\mathbb{A}\mathbf{u}=\mathbf{f}(t,x,\mathbf{u},\Sigma \mathbf{u})&\text{ in }]0,T[\times\Omega,\\\mathbf{u}=\mathbf{0}&\text{ in…

Our purpose of this paper is to study the nonexistence of nonnegative very weak solutions of \begin{equation}\label{eq 0.1} \displaystyle (-\Delta)^\alpha u = u^p+\nu\quad {\rm in}\quad \Omega,\qquad\ u=g\quad {\rm in}\quad \mathbb{…

The Yukawa corrections of order ${\cal O}(\alpha_{ew}m_{t(b)}^2/m_W^2)$, ${\cal O}(\alpha_{ew}m_{t(b)}^3/m_W^3)$ and ${\cal O}(\alpha_{ew}m_{t(b)}^4/m_W^4)$ to the width of sbottom decay into lighter stop plus charged Higgs boson are…

In this paper we discuss the existence and non-existence of weak solutions to parametric fractional equations involving the square root of the Laplacian $A_{1/2}$ in a smooth bounded domain $\Omega\subset \mathbb{R}^n$ ($n\geq 2$) and with…

In this paper we consider a minimization problem of the type $$ I_{\beta,p}(D;\Omega)=\inf\biggl\{\int_\Omega \lvert{D\phi}\rvert^pdx+\beta \int_{\partial^* \Omega}\lvert{\phi}\rvert^pd\mathcal{H}^{n-1},\; \phi \in W^{1,p}(\Omega),\;\phi…

In this paper, we study the following singular nonlinear elliptic problem \begin{equation}\label{eq:1} \left\{ \begin{array}{ll} \displaystyle (-\Delta)^{\frac \alpha 2} u=\lambda |u|^{r-2}u+\mu\frac{|u|^{q-2}u}{|x|^{s}}\quad &{\rm in…

For $\Omega_\e=(0,\e)\times (0,1)$ a thin rectangle, we consider minimization of the two-dimensional nonlocal isoperimetric problem given by \[ \inf_u E^{\gamma}_{\Omega_\e}(u)\] where \[ E^{\gamma}_{\Omega_\e}(u):=…

Quantitative stochastic homogenization of linear elliptic operators is by now well-understood. In this contribution we move forward to the nonlinear setting of monotone operators with $p$-growth. This work is dedicated to a quantitative…

This paper is devoted to the study of the existence of positive solutions for a problem related to a higher order fractional differential equation involving a nonlinear term depending on a fractional differential operator,…

We review the indefinite sublinear elliptic equation $-\Delta u=a(x)u^{q}$ in a smooth bounded domain $\Omega\subset\mathbb{R}^{N}$, with Dirichlet or Neumann homogeneous boundary conditions. Here $0<q<1$ and $a$ is continuous and changes…