Related papers: Bifurcation analysis of the Hardy-Sobolev equation

In this article, we study the following Hardy-Sobolev-Maz'ya type equation: \begin{equation} -\Delta u - \mu \frac{u}{|z|^2} = \frac{|u|^{q-2}u}{|z|^t}, \quad u \in D^{1,2} (\mathbb{R}^n), \end{equation} where $x = (y,z) \in \mathbb{R}^h…

In this paper, we apply a self-similar transformation to convert the parabolic equation with a Sobolev-Hardy term \begin{align*} u_t-\Delta u= \frac{|u|^{q-2}u}{\left|x\right|^s} & \text { in } \mathbb{R}^N \times(0, \infty), \end{align*}…

In this paper, we consider the following problem: $$ (-\Delta)^{s} u -\frac{\zeta u}{|x|^{2s}} = \sum_{i=1}^{k} \frac{|u|^{2^{*}_{s,\theta_{i}}-2}u} {|x|^{\theta_{i}}} , \mathrm{~in~} \mathbb{R}^{N}, $$ where $N\geqslant3$, $s\in(0,1)$,…

In this paper, we consider the weighted fourth order equation $$\Delta(|x|^{-\alpha}\Delta u)+\lambda \text{div}(|x|^{-\alpha-2}\nabla u)+\mu|x|^{-\alpha-4}u=|x|^\beta u^p\quad \text{in} \quad \mathbb{R}^n \backslash \{0\},$$ where $n\geq…

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 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…

Given $N\geq 3,$ we consider the critical Hardy-Sobolev equation $-\Delta u-\frac{\gamma}{|x|^2}u=\frac{|u|^{2^*(s)-2}u}{|x|^s}$ in $\mathbb{R}^N\setminus \{0\},$ where $0<\gamma<\gamma_{H}:=\left(\frac{N-2}{2}\right)^2,\,s\in (0,2)$ and…

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…

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 prove new results concerning the nonlinear scalar field equation \begin{equation*} \left\{ \begin{array}{ll} -\Delta u = g(u)&\quad \hbox{in }\mathbb{R}^N,\; N\geq 3, u\in H^1(\mathbb{R}^N)& \end{array} \right. \end{equation*} with a…

In this paper, we study the existence and multiplicity of solutions for the following fractional problem involving the Hardy potential and concave-convex nonlinearities: $$({-}{ \Delta})^{\frac{\alpha}{2}}u- \gamma \frac{u}{|x|^{\alpha}}=…

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 dimension $N\geq 5$, and for $0<s<4$ with $\gamma\in\mathbb{R}$, we study the existence of nontrivial weak solutions for the doubly critical problem $$\Delta^2 u-\frac{\gamma}{|x|^4}u=…

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…

In this paper, we first prove some propositions of Sobolev spaces defined on a locally finite graph $G=(V,E)$, which are fundamental when dealing with equations on graphs under the variational framework. Then we consider a nonlinear…

In this paper we study the Hardy problem in R^N with N>2 and in a ball B of R^N. Using a suitable map we transform the Hardy problem into another one without the singular term. Then we obtain some bifurcation results from the radial…

Many existence and nonexistence results are known for nonnegative radial solutions $u\in D^{1,2}(\mathbb{R}^{N})\cap L^{2}(\mathbb{R}^{N},\left|x\right| ^{-\alpha }dx)$ to the equation \[ -\triangle u+\dfrac{A}{\left| x\right| ^{\alpha…

We consider the following system of Liouville equations: $$\left\{\begin{array}{ll}-\Delta u_1=2e^{u_1}+\mu e^{u_2}&\text{in }\mathbb R^2\\-\Delta u_2=\mu e^{u_1}+2e^{u_2}&\text{in }\mathbb R^2\\\int_{\mathbb…

In this paper, we study the existence of radial and nonradial solutions to the scalar field equations with fractional operators. For radial solutions, we prove the existence of infinitely many solutions under $N \geq 2$. We also show the…

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…