Related papers: A note on radial solutions to the critical Lane-Em…

For the following Neumann problem in a ball $$\begin{cases} -\Delta_p u+u^{p-1}=u^{q-1}\quad&\text{in }B,\\ u>0,\,u\text{ radial}\quad&\text{in }B,\\ \frac{\partial u}{\partial \nu}=0\quad&\text{on }\partial B, \end{cases}$$ with…

In this paper, we address the solvability of the critical Lane-Emden system \[\begin{cases} -\Delta u=|v|^{p-1}v &\mbox{in } \Omega_\epsilon,\\ -\Delta v=|u|^{q-1}u &\mbox{in } \Omega_\epsilon,\\ u=v=0 &\mbox{on } \partial \Omega_\epsilon,…

We study the following coupled Schr\"odinger system \ds -\Delta u+u=u^{2^*-1}+\be u^{\frac{2^*}{2}-1}v^{\frac{2^*}{2}}+\la_1u^{\al-1}, &x\in \R^N, \ds -\Delta v+v=v^{2^*-1}+\be u^{\frac{2^*}{2}}v^{\frac{2^*}{2}-1}+\la_2v^{r-1}, &x\in \R^N,…

We consider radial solutions of the slightly subcritical problem $-\Delta u_\varepsilon = |u_\varepsilon|^{\frac{4}{n-2}-\varepsilon}u_\varepsilon$ either on $\mathbb R^n$ ($n\geq 3$) or in a ball $B$ satisfying Dirichlet or Neumann…

We prove the existence of positive solutions for the supercritical nonlinear fractional Schr\"odinger equation $(-\Delta)^s u+V(x)u-u^p=0$ in $\mathbb R^n$, with $u(x)\to 0$ as $|x|\to +\infty$, where $p>\frac{n+2s}{n-2s}$ for $s\in (0,1),…

We study entire bounded solutions to the equation $\Delta u - u + u^3 = 0$ in $\mathbb R^2$. Our approach is purely variational and is based on concentration arguments and symmetry considerations. This method allows us to construct in a…

In this paper we prove an existence result to the problem $$\left\{\begin{array}{ll} -\Delta u = |u|^{p-1} u \qquad & \text{in} \Omega, \\ u= 0 & \text{on} \partial\Omega, \end{array} \right. $$ where $\Omega$ is a bounded domain in…

We study the existence of positive increasing radial solutions for superlinear Neumann problems in the ball. We do not impose any growth condition on the nonlinearity at infinity and our assumptions allow for interactions with the spectrum.…

We study the following Lane-Emden system \[ -\Delta u=|v|^{q-1}v \quad \text{ in } \Omega, \qquad -\Delta v=|u|^{p-1}u \quad \text{ in } \Omega, \qquad u_\nu=v_\nu=0 \quad \text{ on } \partial \Omega, \] with $\Omega$ a bounded regular…

Equation $(-\Delta+k^2)u+f(u)=0$ in $D$, $u\mid_{\partial D}=0$, where $k=\const>0$ and $D\subset\R^3$ is a bounded domain, has a solution if $f:\R\to\R$ is a continuous function in the region $|u|\geq a$, piecewise-continuous in the region…

We study Liouville-type theorem for polyharmonic H\'enon-Lane-Emden system $(-\Delta)^mu=|x|^av^p,\; (-\Delta)^mv=|x|^bu^q$ when $m,p,q\geq 1, pq\ne 1$, and $a,b\geq 0$. It is a natural conjecture that the nonexistence of positive solutions…

We investigate the existence and multiplicity of positive solutions to the problem \begin{equation} \begin{cases} \begin{aligned} - \Delta_{\gamma} u &= \lambda u^{p} + u^{-\delta} &\quad \text{in } \Omega, \quad u &= 0 &\quad \text{on }…

This paper is devoted to the study of stable radial solutions of $-\Delta u=f(u) \mbox{ in } \mathbb{R}^N\setminus B_1=\{ x\in \mathbb{R}^N : \vert x\vert\geq 1\}$, where $f\in C^1(\mathbb{R})$ and $N\geq 2$. We prove that such solutions…

We consider weak non-negative solutions to the critical $p$-Laplace equation in $\mathbb{R}^N$, $-\Delta_p u =u^{p^*-1}$ in the singular case $1<p<2$. We prove that if the nonlinearity is locally Lipschitz continuous, namely $p^*\geqslant2$…

We consider the semilinear Lane-Emden problem \begin{equation}\label{problemAbstract} \left\{\begin{array}{lr}-\Delta u= |u|^{p-1}u\qquad \mbox{ in }B u=0\qquad\qquad\qquad\mbox{ on }\partial B \end{array}\right.\tag{$\mathcal E_p$}…

In this paper we investigate the existence of the positive solutions for the following nonlinear Schr\"odinger equation $$ -\triangle u+V(x)u=K(x)|u|^{p-2}u\ {in}\ \mathbb{R}^N $$ where $V(x)\sim a|x|^{-b}$ and $K(x)\sim \mu|x|^{-s}$ as…

This paper deals with existence and multiplicity of positive solutions to the following class of nonlocal equations with critical nonlinearity: \begin{equation} \tag{$\mathcal E$} (-\Delta)^s u = a(x)…

We consider non-negative distributional solutions $u\in C^1 (\bar{B_R } )$ to the equation $-\mbox{div} [g(|\nabla u|)|\nabla u|^{-1} \nabla u ] = f(|x|,u)$ in a ball $B_R$, with $u=0$ on $\partial B_R $, where $f$ is continuous and…

We prove the existence, uniqueness, and sharp bilateral pointwise estimates for positive bounded solutions to the Lane--Emden type problem \[ \begin{cases} L u = \sum\limits_{i=1}^{m}\sigma_{i} u^{q_{i}}+\sigma_0, \quad u\geq0 & \text{in }…

We consider the following slightly supercritical problem for the Lane-Emden system with Neumann boundary conditions: \begin{equation*} \begin{cases} -\Delta u_1=|u_2|^{p_\epsilon-1}u_2,\ &in\ \Omega,\\ -\Delta u_2=|u_1|^{q_\epsilon-1}u_1, \…