Related papers: Multiple solutions for an indefinite elliptic prob…

Let $N \ge 4$, $\Omega$ be a bounded domain in $\mathbb{R}^N$, and let $\Sigma \subset \Omega$ be a smooth closed submanifold of dimension $k$ with $2 \le k \le N-2$. We study the existence of positive solutions $u \in H_0^1(\Omega)$ to the…

Consider the problem \begin{eqnarray*} -\Delta u_\e &=& v_\e^p \quad v_\e>0\quad {in}\quad \Omega, -\Delta v_\e &=& u_\e^{q_\e}\quad u_\e>0\quad {in}\quad \Omega, u_\e&=&v_\e\:\:=\:\:0 \quad {on}\quad \partial \Omega, \end{eqnarray*} where…

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 existence of solutions to a nonlinear degenerate elliptic equation of the form \[ \begin{cases} -\Delta_{1} u+ \frac{|D u|}{(1-u)^{\gamma}}=g & \mbox{in $\Omega$,}\\ u=0 \hfill & \mbox{on $\partial\Omega$,} \end{cases} \] in a…

We are interested in the following Dirichlet problem $$ \left\{ \begin{array}{ll} -\Delta u + \lambda u - \mu \frac{u}{|x|^2} - \nu \frac{u}{\mathrm{dist}\,(x,\mathbb{R}^N \setminus \Omega)^2} = f(x,u) & \quad \mbox{in } \Omega \\ u = 0 &…

We deal with existence, uniqueness and regularity of nonnegative solutions to a Dirichlet problem for equations as \begin{equation*} \displaystyle -\operatorname{div}\left(\frac{|\nabla u|^{p-2}\nabla u}{(1+u)^{\theta(p-1)}}\right) = h(u)f…

We consider the existence and nonexistence of positive solution for the following Br\'ezis-Nirenberg problem with logarithmic perturbation: \begin{equation*} \begin{cases} -\Delta u={\left|u\right|}^{{2}^{\ast }-2}u+\lambda u+\mu u\log…

We study the following nonlinear and nonlocal elliptic equation in~$\R^n$ $$ (-\Delta)^s u = \epsilon\,h\,u^q + u^p \ {\mbox{ in }}\R^n, $$ where~$s\in(0,1)$, $n>2s$, $\epsilon>0$ is a small parameter, $p=\frac{n+2s}{n-2s}$, $q\in(0,1)$,…

We investigate the following fractional $p$-Laplacian equation \[ \begin{cases} \begin{aligned} (-\Delta)_p^s u&=\lambda |u|^{q-2}u+|u|^{p_s^*-2}u &&\text{in}~\Omega,\\ u &=0 &&\text{in}~ \mathbb{R}^n\setminus\Omega, \end{aligned}…

We investigate the problem $$-\Delta u = \lambda b(x)|u|^{q-2}u +a(x)|u|^{p-2}u \mbox{ in } \Omega, \quad \frac{\partial u}{\partial \mathbf{n}} = 0 \mbox{ on } \partial \Omega, \leqno{(P_\lambda)} $$ where $\Omega$ is a bounded smooth…

In this article, we investigate the existence and multiplicity of solutions to the Robin problem \begin{equation*} \begin{cases} -\Delta u = \lambda f(u) & \text{in } \Omega, \frac{\partial u}{\partial \nu} + \gamma u=0 & \text{on }…

Let $\Omega \subset \mathbb{R}^N$ ($N \geq 3$) be a $C^2$ bounded domain and $\Sigma \subset \Omega$ is a $C^2$ compact boundaryless submanifold in $\mathbb{R}^N$ of dimension $k$, $0\leq k < N-2$. For $\mu\leq (\frac{N-k-2}{2})^2$, put…

Let $\Omega\subset\mathbb{C}$ be a bounded domain. In this note, we use complex variable methods to study the number of critical points of the function $v=v_\Omega$ that solves the elliptic problem $\Delta v = -2$ in $\Omega,$ with boundary…

We prove that if the elliptic problem $-\Delta u+b(x)|\nabla u|=c(x)u$ with $c\ge0$ has a positive supersolution in a domain $\Omega$ of $ \IR^{N\ge 3}$, then $c,b$ must satisfy the inequality \[\sqrt{ \int_\Omega c\phi^2}\le \sqrt{…

We investigate the existence and multiplicity of solutions to the following $p(x)$-Laplacian problem in $\mathbb{R}^{N}$ via critical point theory \begin{equation*} \left\{ \begin{array}{l} -\bigtriangleup _{p(x)}u+V(x)\left\vert…

We study the fractional laplacian problem (-\Delta)^s u &=& u^p -\epsilon u^q \quad\text{in }\quad \Omega, u &\in& H^s(\Omega)\cap L^{q+1}(\Omega),u &>&0 \quad\text{in }\quad \Omega, u&=&0 \quad\text{in}\quad \mathbb{R}^N\setminus\Omega,…

We study positive solutions of equation (E) $-\Delta u + u^p|\nabla u|^q= 0$ ($0<p$, $0\leq q\leq 2$, $p+q>1$) and other related equations in a smooth bounded domain $\Omega \subset {\mathbb R}^N$. We show that if $N(p+q-1)<p+1$ then, for…

We consider an elliptic differential inequality: $\vert \Delta u(x) \vert \le C_0(\YYYY^{-\gamma}\vert u(x)\vert + \YYYY^{-\theta}\vert \nabla u(x)\vert)$ in an exterior domain $\R^n \setminus \ooo{U}$, where $U$ is a simply connected…

On a closed Riemannian manifold $(M^n ,g)$ with a proper isoparametric function $f$ we consider the equation $\Delta^2 u -\alpha \Delta u +\beta u = u^q$, where $\alpha$ and $\beta$ are positive constants satisfying that $\alpha^2 \geq 4…

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,…