Related papers: Nonlocal problems at nearly critical growth

In this paper we are concerned with the number of critical points of solutions of nonlinear elliptic equations. We will deal with the case of non-convex, contractile and non-contractile planar domains. We will prove results on the estimate…

In this paper we study the following problem \begin{equation} \begin{cases} -\Delta u=f(u)~&\mbox{in}\ \Omega_\varepsilon,\\ u>0~&\mbox{in}\ \Omega_\varepsilon,\\ u=0~&\mbox{on}\ \partial\Omega_\varepsilon, \end{cases} \end{equation} where…

We investigate asymptotic behavior of solutions for nonlocal elliptic boundary value problems in plane angles and in ${\mathbb R}^2\backslash\{0\}$. Such problems arise as model ones when studying asymptotics of solutions for nonlocal…

We consider the nonlinear and nonlocal problem $$ A_{1/2}u=|u|^{2^\sharp-2}u\ \text{in \Omega, \quad u=0 \text{on} \partial\Omega $$where $A_{1/2}$ represents the square root of the Laplacian in a bounded domain with zero Dirichlet boundary…

In this paper, we consider a biharmonic equation under the Navier boundary condition and with a nearly critical exponent $(P_\epsilon): \Delta^2u=u^{9-\epsilon}, u>0$ in $\Omega$ and $u=\Delta u=0$ on $\partial\Omega$, where $\Omega$ is a…

We study asymptotic behaviour of positive ground state solutions of the nonlinear Choquard equation $$ -\Delta u+\varepsilon u=(I_\alpha \ast |u|^{p})|u|^{p-2}u+ |u|^{q-2}u \quad {\rm in} \ \mathbb R^N, $$ where $N\ge 3$ is an integer,…

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…

We prove the existence of infinitely many nonnegative solutions to the following nonlocal elliptic partial differential equation involving singularities \begin{align} (-\Delta)_{p(\cdot)}^{s}…

Let $s\in(0,1),$ $1<p<\frac{N}{s}$ and $\Omega\subset\mathbb{R}^N$ be an open bounded set. In this work we study the existence of solutions to problems ($E_\pm$) $Lu\pm g(u)=\mu$ and $u=0$ a.e. in $\mathbb{R}^N\setminus\Omega,$ where $g\in…

We study the asymptotic profile, as $\hbar\rightarrow 0$, of positive solutions to $$-\hbar^2\Delta u+V(x)u-\hbar^{2+\gamma}u\Delta u^2=K(x)|u|^{p-2}u,\ \ x\in \mathbb{R}^N $$ where $\gamma\geq 0$ is a parameter with relevant physical…

We study the following nonlinear Hartree-type equation \begin{equation*} -\Delta u+V(x)u-a(\frac{1}{|x|^\gamma}\ast |u|^2)u=\lambda u,~\text{in}~\mathbb{R}^N, \end{equation*} where $a>0$, $N\geq3$, $\gamma\in(0,2)$ and $V(x)$ is an external…

Let $\Omega$ be a smooth, bounded domain of $\mathbb{R}^{N}$, $\omega$ be a positive, $L^{1}$-normalized function, and $0<s<1<p.$ We study the asymptotic behavior, as $p\rightarrow\infty,$ of the pair $\left( \sqrt[p]{\Lambda_{p}%…

We establish the existence of finitely many sign-changing solutions to the Lane-Emden system $$-\Delta u=|v|^{q-2}v,\quad -\Delta v=|u|^{p-2}u \quad \text{ in }\mathbb{R}^N, \ \ N\geq 4,$$ where the exponents $p$ and $q$ lie on the critical…

We study existence and multiplicity of nontrivial solutions of the following problem $$ \left\{ \begin{array}{rcll} -\Delta_p u+(-\Delta_p)^{s} u & = & \lambda|u|^{q-2}u+|u|^{p^{\ast}-2}u & \mbox{ in }\Omega,\\ u & = & 0 & \mbox{ on }…

We study the fractional Schr\"odinger equations with a vanishing parameter: $$ (-\Delta)^s u+u =|u|^{p-2}u+\lambda|u|^{q-2}u \text{ in }\mathbb{R}^N,\quad u \in H^s(\mathbb{R}^N),$$ where $s\in(0,1)$, $N>2s$, $2<q<p\leq…

This work is devoted to the study of the nonlinear second-order neutral difference equations with quasi-differences of the form $$ \Delta \left( r_{n} \Delta \left( x_{n}+q_{n}x_{n-\tau}\right)\right)= a_{n}f(x_{n-\sigma})+b_n%, \ n\geq n_0…

In this paper we consider semilinear equations $-\Delta u=f(u)$ with Dirichlet boundary conditions on certain convex domains of the two dimensional model spaces of constant curvature. We prove that a positive, semi-stable solution $u$ has…

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 large-time behaviour of the solutions of the evolution equation involving nonlinear diffusion and gradient absorption, $$ \partial_t u - \Delta_p u + |\nabla u|^q=0 . $$ We consider the problem posed for $x\in \real^N$ and t>0…

In this paper, the asymptotic behavior of the solutions of a monotone problem posed in a locally periodic oscillating domain is studied. Nonlinear monotone boundary conditions are imposed on the oscillating part of the boundary whereas the…