Related papers: Bubbling solutions for supercritical problems on m…

In this work we study the existence of nodal solutions for the problem $$ -\Delta u = \lambda u e^{u^2+|u|^p} \text{ in }\Omega, \; u = 0 \text{ on }\partial \Omega, $$ where $\Omega\subseteq \mathbb R^2$ is a bounded smooth domain and…

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 =…

We consider the equation $- \e^2 \D u + u= u^p$ in $\Omega \subseteq \R^N$, where $\Omega$ is open, smooth and bounded, and we prove concentration of solutions along $k$-dimensional minimal submanifolds of $\partial \O$, for $N \geq 3$ and…

We construct positive solutions of the semilinear elliptic problem $\Delta u+ \lambda u + u^p = 0$ with Dirichet boundary conditions, in a bounded smooth domain $\Omega \subset \R^N$ $(N\geq 4)$, when the exponent $p$ is supercritical and…

Let $\Omega$ be a bounded domain in $\mathbb{R}^2$ with smooth boundary, we study the following elliptic Dirichlet problem $$ \begin{cases} -\Delta\upsilon= e^{\upsilon}-s\phi_1-4\pi\alpha\delta_p-h(x)\,\,\,\,…

We consider the slightly subcritical elliptic problem with Hardy term $$ \left\{ \begin{aligned} -\Delta u-\mu\frac{u}{|x|^2} &= |u|^{2^{\ast}-2-\epsilon}u &&\quad \text{in } \Omega\subset\mathbb{R}^N, \\\ u &= 0&&\quad \text{on } \partial…

We prove existence of solutions to the following problem \begin{equation*} \begin{cases} -\Delta_1 u +g(u)|Du|=h(u)f & \text{in $\Omega$,} \\ u=0 & \text{on $\partial\Omega$,} \end{cases} \end{equation*} where $\Omega \subset \mathbb{R}^N$,…

Via a constrained minimization, we find a solution $(\lambda,u)$ to the problem \begin{equation*} \begin{cases} (-\Delta)^m u+\frac{\mu}{|x|^{2m}}u + \lambda u = \eta u^3 + g(u)\\ \int_{\mathbb{R}^{2m}} u^2 \, dx = \rho \end{cases}…

Motivated by the prescribing scalar curvature problem, we study the equation $\Delta_g u +Ku^p=0 (1+\zeta \leq p \leq \frac{n+2}{n-2})$ on locally conformally flat manifolds $(M,g)$ with $R(g)=0$. We prove that when $K$ satisfies certain…

In this paper we study the Lane-Emden-Fowler equation $$(P)_\epsilon\ \{\Delta u+|u|^{q-2}u=0 \ \hbox{in}\ \mathcal D_\epsilon, u=0 \ \hbox{on}\ \partial\mathcal D_\epsilon.$$ Here $\mathcal D_\epsilon = \mathcal D \setminus \{x \in…

We are interested in the structure of the positive radial solutions of the supercritical Neumann problem $\varepsilon^2\Delta u-u+u^p=0$ on a unit ball in $\mathbb{R}^N$ , where $N$ is the spatial dimension and $p>p_S:=(N+2)/(N-2)$, $N\ge…

We consider the problem $\Delta u + \lambda u +u^5 = 0$, $u>0$, in a smooth bounded domain $\Omega$ in ${\mathbb R}^3$, under zero Dirichlet boundary conditions. We obtain solutions to this problem exhibiting multiple bubbling behavior at…

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 construct solutions to a Yamabe type problem on a Riemannian manifold M without boundary and of dimension greater than 2, with nonlinearity close to higher critical Sobolev exponents. These solutions concentrate their mass around a non…

On a closed Riemannian manifold $(M^n ,g)$, we consider the Yamabe-type equation $-\Delta_g u + \lambda u = \lambda |u|^{q-1}u$, where $\lambda \in \mathbb{R}_{+}$ and $q>1$. We assume that $M$ admits a proper isoparametric function $f$…

We consider the problem $-\Delta u+\lambda u=u^{p-1}$, where $u\in H^1_0(\Omega)$ verifies $\|u\|_{L^2}=m>0$, and $\lambda\in [0,+\infty)$. Here, $\mathbb{R}^N\setminus\Omega$ is nonempty and compact. We prove the existence of a solution…

Let (M,g) be a smooth compact, n dimensional Riemannian manifold,with smooth n-1 dimensional boundary. We prove that the stable critical points of the mean curvature of the boundary generates solutions for a singularly perturbed elliptic…

We consider the following Ambrosetti-Prodi type problem \begin{equation} \left\{\begin{array}{ll} -\mathrm{div} (A(x)\nabla u)=|u|^p-t\mathbf{\Psi}(x), &\mbox{in $\Omega$,} \\ u=0, & \mbox{on $\partial \Omega$}, \end{array} \right.…

We consider the fourth-order nonlinear elliptic problem: \begin{equation*} \begin{array}{ll} \Delta(a(x)\Delta u) = a(x) \left\vert u \right\vert^{p-2-\epsilon} u \ \text{ in } \ \Omega, \hspace{0.6cm} u = 0 \ \text{ on } \ \partial \Omega,…

For $N\geq 4$, we let $\Omega$ to be a smooth bounded domain of $\mathbb{R}^N$, $\Gamma$ a smooth closed submanifold of $\Omega$ of dimension $k$ with $1\leq k \leq N-2$ and $h$ a continuous function defined on $\Omega$. We denote by…