Related papers: A nonradial bifurcation result with applications t…

In this paper we consider the problem $$ {ll} -\Delta u=(N+\a)(N-2)|x|^{\a}u^\frac{N+2+2\a}{N-2} & in R^N u>0& in R^N u\in D^{1,2}(R^N). $$ where $N\ge3$. From the characterization of the solutions of the linearized operator, we deduce the…

We consider the H\'enon problem \begin{equation*} \left\{ \begin{array} - - \Delta u = |x|^{\alpha} u^{\frac{N+2+2\alpha}{N-2}-\varepsilon} & \ \ \text{in} \ B_1, \\ u > 0 & \ \ \text{in} \ B_1, \\ u=0 & \ \ \text{on} \ \partial B_1,…

In this paper we obtain, for a semilinear elliptic problem in R^N, families of solutions bifurcating from the bottom of the spectrum of $-\Delta$. The problem is variational in nature and we apply a nonlinear reduction method which allows…

We prove the existence of non-radial entire solution to $$\Delta^2 u+u^{-q}=0\quad\text{in }\mathbb{R}^3,\quad u>0,$$ for $q>1$. This answers an open question raised by P. J. McKenna and W. Reichel (E. J. D. E. \textbf{37} (2003) 1-13).

In this paper, we prove a global bifurcation result for the existence of non-radial branches of solutions to the paramterized family of $\Gamma$-symmetric problems $-\Delta u=f(\alpha,z,u)$, $u|_{\partial D}=0$ on the unit disc $D:=\{z\in…

We obtain an improved version of a recent result concerning the existence of nonnegative nonradial solutions $u\in D^{1,2}(\mathbb{R}^{N})\cap L^{2}(\mathbb{R}^{N},\left| x\right| ^{-\alpha }dx)$ to the equation \[ -\triangle…

We study the following nonlinear scalar field equation $$ -\Delta u=f(u)-\mu u, \quad u \in H^1(\mathbb{R}^N) \quad \text{with} \quad \|u\|^2_{L^2(\mathbb{R}^N)}=m. $$ Here $f\in C(\mathbb{R},\mathbb{R})$, $m>0$ is a given constant and…

In this paper, we study the existence of solution for the following class of nonlocal problem, $$ \left\{ \begin{array}{lcl} -\Delta u=\left(\lambda f(x)-\int_{\R^N}K(x,y)|u(y)|^{\gamma}dy\right)u,\quad \mbox{in} \quad \R^{N}, \\…

In this paper we establish existence of radial and nonradial solutions to the system $$ \begin{array}{ll} -\Delta u_1 = F_1(u_1,u_2) &\text{in }\mathbb{R}^N,\newline -\Delta u_2 = F_2(u_1,u_2) &\text{in }\mathbb{R}^N,\newline u_1\geq 0,\…

We prove the existence of an infinite sequence of distinct non-radial nodal $G-$invariant solutions for the following critical nonlinear elliptic problem: $({\mathrm{P}})\quad {*{20}c} {-\Delta u = |u|^{4/(n-2)}u},\quad u\in…

In this paper, we prove existence of multiple non-radial solutions to the Hardy-Sobolev equation $$\begin{cases} -\Delta u-\displaystyle\frac \gamma{|x|^2}u=\displaystyle\frac{1}{|x|^s}|u|^{p_s-2}u & \text{ in }…

This paper presents local and global bifurcation results for radially symmetric solutions of the cubic Helmholtz system \begin{equation*} \begin{cases} -\Delta u - \mu u = \left( u^2 + b \: v^2 \right) u &\text{ on } \mathbb{R}^3, \\…

This paper is devoted to the study of semi-stable radial solutions $u\notin H^1(B_1)$ of $-\Delta u=f(u) \mbox{in} \overline{B_1}\setminus \{0\}=\{x\in \mathbb{R}^N : 0<\vert x\vert\leq 1\}$, where $f\in C^1(\mathbb{R})$ and $N\geq 2$. We…

By a combination of variational and topological techniques in the presence of invariant cones, we detect a new type of positive axially symmetric solutions of the Dirichlet problem for the elliptic equation $$ -\Delta u + u = a(x)|u|^{p-2}u…

For the equation (-\Delta u = | |x|-2 |^\alpha u^{p-1}), (1 < |x| < 3), we prove the existence of two solutions for (\alpha) large, and of two additional solutions when (p) is close to the critical Sobolev exponent (2^*=2N/(N-2)). A…

We develop numerical algorithms to approximate positive solutions of elliptic boundary value problems with superlinear subcritical nonlinearity on the boundary of the form $-\Delta u + u = 0$ in $\Omega$ with $\frac{\partial u}{\partial…

In this paper we study the Neumann problem\begin{equation*}\begin{cases}-\Delta u+u=u^p \& \text{ in }B\_1 \\u \textgreater{} 0, \& \text{ in }B\_1 \\\partial\_\nu u=0 \& \text{ on } \partial B\_1,\end{cases}\end{equation*}and we show the…

We study the critical Neumann problem \begin{equation*} \begin{cases} -\Delta u = |u|^{2^*-2}u &\text{in }\Sigma_\omega,\\ \quad\frac{\partial u}{\partial\nu}=0 &\text{on }\partial\Sigma_\omega, \end{cases} \end{equation*} in the unbounded…

In this paper, we build infinitely many non-radial sign-changing solutions to the critical problem: \begin{equation*} \left\{\begin{array}{rlll} -\Delta u&=|u|^{\frac{4}{N-2}}u, &\hbox{ in }\Omega,\\ u&=0, &\hbox{ on }\partial\Omega.…

In this paper we study the problem -\Delta u =\left(\frac{2+\alpha}{2}\right)^2\abs{x}^{\alpha}f(\lambda,u), & \hbox{in}B_1 \\ u > 0, & \hbox{in}B_1 u = 0, & \hbox{on} \partial B_1 where $B_1$ is the unit ball of $\R^2$, $f$ is a smooth…