Related papers: On a nonlinear Robin problem with an absorption te…

We study nonnegative solutions of the boundary value problem $$-\Delta u = \lambda c(x)u + \mu(x)|\nabla u|^2 + h(x),\quad u\in H^1_0(\Omega)\cap L^\infty(\Omega), \leqno(P_\lambda)$$ where $\Omega$ is a smooth bounded domain, $\mu, c\in…

Here we study the nonnegative solutions of the viscous Hamilton-Jacobi problem \[ \left\{\begin{array} [c]{c}% u_{t}-\nu\Delta u+|\nabla u|^{q}=0, u(0)=u_{0}, \end{array} \right. \] in $Q_{\Omega,T}=\Omega\times\left(0,T\right) ,$ where…

We consider nonlinear second order elliptic problems of the type \[ -\Delta u=f(u) \text{ in } \Omega, \qquad u=0 \text{ on } \partial \Omega, \] where $\Omega$ is an open $C^{1,1}$-domain in $\mathbb{R}^N$, $N\geq 2$, under some general…

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…

Let $\Omega \subset \mathbb{R}^n$ be a bounded smooth domain (open and connected) in $\mathbb{R}^n$. Given $u_0\in L^2(\Omega)$, $g\in L^\infty(\Omega)$ and $\lambda \in \mathbb{R}$, our purpose is to describe the asymptotic behavior of…

We analyze nonnegative solutions of the nonlinear elliptic problem $\Delta u=\frac{\lambda f(x)}{u^2}+P$, where $\lambda>0$ and $P\geq0$, on a bounded domain $\Omega$ of $\mathbb{R}^N$ ($N\geq 1$) with a Dirichlet boundary condition. This…

Let $\Omega$ be a bounded open interval, and let $p>1$ and $q\in\left(0,p-1\right) $. Let $m\in L^{p^{\prime}}\left(\Omega\right) $ and $0\leq c\in L^{\infty}\left(\Omega\right) $. We study existence of strictly positive solutions for…

We prove the existence of a solution to a singular anisotropic elliptic equation in a bounded open subset $\Omega$ of $\mathbb R^N$ with $N\ge 2$, subject to a homogeneous boundary condition: \begin{equation} \label{eq0} \left\{…

This paper considers a local and non-local problem characterized by singular nonlinearity and a source term. Specifically, we focus on the following problem: \begin{equation}\label{A}\tag{P} -\Delta_{p} u + (-\Delta)^{s}_{q} u = f(x)…

We establish the uniqueness of the positive solution for equations of the form $-\Delta u=au-b(x)f(u)$ in $\Omega$, $u|\_{\partial\Omega}=\infty$. The special feature is to consider nonlinearities $f$ whose variation at infinity is…

We are devoted to the study of the following nonlinear $p$-Laplacian Schr\"odinger equation with $L^{p}$-norm constraint \begin{align*} \begin{cases} &-\Delta_{p} u=\lambda |u|^{p-2}u +|u|^{r-2}u\quad\mbox{in}\quad\Omega,\\…

By applying Ricceri's variational principle, we demonstrate the existence of solutions for the following Robin problem \begin{equation*}\left\{ \begin{array}{cc}-\func{div}\left( \omega _{1}(x)\left\vert \nabla u\right\vert^{p(x)-2}\nabla…

We consider the $2m$-th order elliptic boundary value problem $Lu=f(x,u)$ on a bounded smooth domain $\Omega\subset\R^N$ with Dirichlet boundary conditions on $\partial\Omega$. The operator $L$ is a uniformly elliptic linear operator of…

We study the semilinear indefinite elliptic problem \[ -\Delta u = Q_\Omega |u|^{p-2}u \quad \text{in } \mathbb{R}^N, \] where $Q_\Omega = \chi_\Omega - \chi_{\mathbb{R}^N \setminus \Omega}$, $\Omega \subset \mathbb{R}^N$ is a bounded…

We study perturbations of the eigenvalue problem for the negative Laplacian plus an indefinite and unbounded potential and Robin boundary condition. First we consider the case of a sublinear perturbation and then of a superlinear…

We consider in this paper the nonlinear elliptic equation with Neumann boundary condition \begin{align*} \begin{cases} \Delta u=a|u|^{m-1}u\,\,\mbox{ in }\,\,\rnp\\ \dfrac{\partial u}{\partial t}=b|u|^{\eta-1}u+f\,\,\mbox{ on…

In this paper, we study the existence of a solution for a class of Dirichlet problems with a singularity and a convection term. Precisely, we consider the existence of a positive solution to the Dirichlet problem $$-\Delta_p u =…

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…

We study the generalized boundary value problem for nonnegative solutions of of $-\Delta u+g(u)=0$ in a bounded Lipschitz domain $\Omega$, when $g$ is continuous and nondecreasing. Using the harmonic measure of $\Omega$, we define a trace…

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…