Related papers: A scalar field equation on hyperbolic space with i…

We study the existence and non-existence of positive solutions for the following class of nonlinear elliptic problems in the hyperbolic space $$ -\Delta_{\mathbb{B}^N} u-\lambda u=a(x)u^{p-1} \, + \, \varepsilon u^{2^*-1}…

We study the existence of positive solutions for the following class of scalar field problem on the hyperbolic space $$ -\Delta_{\mathbb{H}^N} u - \lambda u = a(x) |u|^{p-1} \, u\;\;\text{in}\;\mathbb{B}^{N}, \quad u \in…

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

In this paper, we consider systems of semilinear elliptic equations \displaystyle -\Delta_{\mathbb{H}^{N}}u=|v|^{p-1}v, \displaystyle -\Delta_{\mathbb{H}^{N}}v=|u|^{q-1}u, in the whole of Hyperbolic space $\mathbb{H}^{N}$. We establish…

This article investigates the multiplicity of solutions to the Brezis-Nirenberg problem on smooth bounded domains in the hyperbolic space $\mathbb{B}^N$ for $N \ge 4$. Specifically, we study the critical semilinear equation…

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…

Semilinear elliptic equations of the form $-\Delta u =\lambda|u|^{p-2}u- |u|^{q-2}u$ in bounded and unbounded domains are considered. In the plane of exponents $p\times q$, the so-called critical exponents curve is introduced which…

This paper addresses the following problem. \begin{equation} \left\{ \begin{array}{lr} -{\Delta}u=\lambda I_\alpha*_\Omega u+|u|^{2^*-2}u\mbox{ in }\Omega ,\nonumber u\in H_0^1(\Omega).\nonumber \end{array} \right. \end{equation} Here,…

In this paper, we study the existence of solutions for the following superlinear elliptic equation with nonlinear boundary value condition $$ \left\{ \begin{array}{ll} -\Delta u+u=|u|^{r-2}u &\text{in} \; \Omega,\\ \\ \frac{\partial…

We study the following semilinear biharmonic equation $$ \left\{\begin{array}{lllllll} \Delta^{2}u=\frac{\lambda}{1-u}, &\quad \mbox{in}\quad \B, u=\frac{\partial u}{\partial n}=0, &\quad \mbox{on}\quad \partial\B, \end{array} \right.…

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…

We prove that all positive solutions of $-\Delta u = u^{\frac{2n}{n-2}}$ on the upper half space $\mathbb{R}^n_{+}$ (for $n \geq 3$) satisfying the boundary condition $D_{x_n}u = -u^{\frac{n}{n-2}}$ are of the form $u(x) = a \left(…

In 2012, Y.Y. Li and C.-S. Lin (Arch. Ration. Mech. Anal., 203(3): 943-968) posed an open problem concerning the existence of positive solutions to the elliptic equation $$ \begin{cases} -\Delta u = -\lambda |x|^{-s_1}|u|^{p-2}u +…

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…

In this paper we shall classify all positive solutions of $ \Delta u =a u^p$ on the upper half space $ H =\Bbb{R}_+^n$ with nonlinear boundary condition $ {\partial u}/{\partial t}= - b u^q $ on $\partial H$ for both positive parameters $a,…

We look for nonconstant, positive, radially nondecreasing solutions of the quasilinear equation $-\Delta_p u+u^{p-1}=f(u)$ with $p>2$, in the unit ball $B$ of $\mathbb R^N$, subject to homogeneous Neumann boundary conditions. The…

We verify the existence of radial positive solutions for the semi-linear equation $$ -\,\Delta u=u^{p}\,-\,V(y)\,u^{q},\,\quad\quad u>0,\quad\quad\mbox{ in }\mathbb{R}^N$$ where $N\geq 3$, $p$ is close to $p^*:=(N+2)/(N-2)$, and $V$ is a…

The purpose of this paper is to study nonlinear singular parabolic equations with $p(x)$- Laplacian. Precisely, we consider the following problem and discuss the existence of a non-negative weak solution. \begin{align*} \frac{\partial…

In this paper we deal with the equation \[-\Delta_p u+|u|^{p-2}u=|u|^{q-2}u\] for $1<p<2$ and $q>p$, under Neumann boundary conditions in the unit ball of $\mathbb R^N$. We focus on the three positive, radial, and radially non-decreasing…

In this paper, we consider the logistic elliptic equation $-\Delta u = u- u^{p}$ in a smooth bounded domain $\Omega \subset \mathbb{R}^{N}$, $N\geq2$, equipped with the sublinear Neumann boundary condition $\frac{\partial u}{\partial \nu} =…