Related papers: A nonlinear parabolic problem with singular terms …

The objective of this work is to investigate a nonlocal problem involving singular and critical nonlinearities:\begin{equation*}\left\{\begin{array}{ll} ([u]_{s,p}^p)^{\sigma-1}(-\Delta)^s_p u = \frac{\lambda}{u^{\gamma}}+u^{ p_s^{*}-1…

We study the existence and uniqueness of the positive solutions of the problem (P): $\partial_tu-\Delta u+u^q=0$ ($q>1$) in $\Omega\times (0,\infty)$, $u=\infty$ on $\partial\Omega\times (0,\infty)$ and $u(.,0)\in L^1(\Omega)$, when…

We study the nonlocal nonlinear problem \begin{equation}\label{ppp} \left\{ \begin{array}[c]{lll} (-\Delta)^s u = \lambda f(u) & \mbox{in }\Omega, \\ u=0&\mbox{on } \mathbb{R}^N\setminus\Omega, \end{array} \right. \tag{$P_{\lambda}$}…

The aim goal of this paper is to treat the following problem \begin{equation*} \left\{ \begin{array}{rcll} u_t+(-\D^s_{p}) u &=&\dyle \l \dfrac{u^{p-1}}{|x|^{ps}} & \text{ in } \O_{T}=\Omega \times (0,T), \\ u&\ge & 0 & \text{ in }\ren…

Let $b_{\alpha}^{p}(\mathbb{R}^{1+n}_{+})$ be the space of solutions to the parabolic equation $\partial_{t}u+(-\triangle)^{\alpha}u=0$ $(\alpha\in(0, 1])$ having finite $L^{p}(\mathbb{R}^{1+n}_{+})$ norm. We characterize nonnegative Radon…

We construct multibump nodal solutions of the elliptic equation $$ -\Delta u=a^+[\lambda u+ f(\, \cdot\,, u)]-\mu a^- g(\, \cdot\,, u) $$ in $H^1_0(\Omega)$, when $\mu$ is large, under appropriate assumptions, for $f$ superlinear and…

In a cylinder $\Omega_T=\Omega\times (0,T)\subset \R^{n+1}_+$ we study the boundary behavior of nonnegative solutions of second order parabolic equations of the form \[ Hu =\sum_{i,j=1}^ma_{ij}(x,t) X_iX_ju - \p_tu = 0, \…

We consider the boundary value problem $-\Delta_p u_\lambda -\Delta_q u_\lambda =\lambda g(x) u_\lambda^{-\beta}$ in $\Omega$ , $u_\lambda=0$ on $\partial \Omega$ with $u_\lambda>0$ in $\Omega.$ We assume $\Omega$ is a bounded open set in…

If $\Omega$ is a bounded domain in $\mathbb R^N$ and $f$ a continuous increasing function satisfying a super linear growth condition at infinity, we study the existence and uniqueness of solutions for the problem (P): $\partial_tu-\Delta…

We study the periodic boundary value problem associated with the second order nonlinear differential equation $$ u" + c u' + \left(a^{+}(t) - \mu \, a^{-}(t)\right) g(u) = 0, $$ where $g(u)$ has superlinear growth at zero and at infinity,…

In this paper, we investigate the existence and uniqueness of solutions for the following model problem, involving singularities and inhomogeneous Robin boundary conditions \begin{equation*} \left\{ \begin{array}{ll}…

We proceed with the investigation of the problem $(P_\lambda): $ $-\Delta u = \lambda b(x)|u|^{q-2}u +a(x)|u|^{p-2}u \ \mbox{ in } \Omega, \ \ \frac{\partial u}{\partial \mathbf{n}} = 0 \ \mbox{ on } \partial \Omega$, where $\Omega$ is a…

We study the periodic boundary value problem associated with the second order nonlinear equation \begin{equation*} u'' + ( \lambda a^{+}(t) - \mu a^{-}(t) ) g(u) = 0, \end{equation*} where $g(u)$ has superlinear growth at zero and sublinear…

In this paper we prove the existence of at least one positive solution for the nonlocal semipositone problem \[ \displaystyle \left\{\begin{array}{rcll} (-\Delta)_p^s(u) &=& \lambda f(u) \qquad & \text{in} \ \ \Omega \\u &=& 0 & \text{in} \…

Let $\Omega \subset {\mathbb R}^N$ ($N \geq 3$) be a $C^2$ bounded domain and $F \subset \partial \Omega$ be a $C^2$ submanifold of dimension $0 \leq k \leq N-2$. Put $\delta_F(x)=dist(x,F)$, $V=\delta_F^{-2}$ in $\Omega$ and $L_{\gamma…

The aim of this paper is to give global nonexistence and blow--up results for the problem $$ \begin{cases} u_{tt}-\Delta u+P(x,u_t)=f(x,u) \qquad &\text{in $(0,\infty)\times\Omega$,}\\ u=0 &\text{on $(0,\infty)\times \Gamma_0$,}\\…

This paper is concerned with the study of multiple positive solutions to the following elliptic problem involving a nonhomogeneous operator with nonstandard growth of $p$-$q$ type and singular nonlinearities \begin{equation*} \left\{…

We consider the problem: $$\inf_{{u}\in {H}^{1}_{g}(\Omega),\|u\|_{q}=1} \int_{\Omega}{p(x)}|\nabla{u(x)}|^{2}dx-\lambda\int_{\Omega}| u(x)|^{2}dx$$ where $\Omega$ is a bounded domain in $\R^{n}$, ${n}\geq{4}$, $ p :…

Let $\Omega \subset\mathbb{R}^N$ ($N\geq 3$) be a $C^2$ bounded domain and $\Sigma \subset \partial\Omega$ be a $C^2$ compact submanifold without boundary, of dimension $k$, $0\leq k \leq N-1$. We assume that $\Sigma = \{0\}$ if $k = 0$ and…

In this work, we show existence and uniqueness of positive solutions of $H(Du, D^2u)+\chi(t)|Du|^\Gamma-f(u)u_t=$ in $\Omega\times(0, T)$ and $u=h$ on its parabolic boundary. The operator $H$ satisfies certain homogeneity conditions,…