Related papers: Uniqueness of radial solutions for $m$-Laplacian e…

We prove general uniqueness results for radial solutions of linear and nonlinear equations involving the fractional Laplacian $(-\Delta)^s$ with $s \in (0,1)$ for any space dimensions $N \geq 1$. By extending a monotonicity formula found by…

In this paper, we consider radial distributional solutions of the quasilinear equation $-\Delta_N u=f(u)$ in the punctured open ball $ B_R\backslash\{0\}\subset \RR^N$, $N \geq 2$. We obtain sharp conditions on the nonlinearity $f$ for…

We study radial viscosity solutions to the equation \[ -\ |Du\ |^{q-2}\Delta_{p}^{N}u=f(\ |x\ |)\quad\text{in }B_{R}\subset\mathbb{R}^{N}, \] where $f\in C[0,R)$, $p,q\in(1,\infty)$ and $N\geq2$. Our main result is that $u(x)=v(\ |x\ |)$ is…

In 2013, R.L. Frank and E. Lenzmann [R.L. Frank, E. Lenzmann, Uniqueness of non-linear ground states for fractional Laplacians in $\mathbb{R}$, Acta Math. 210 (2) (2013) 261-318] study the following problem: \begin{align*} (-\Delta)^su + u…

We establish the uniqueness of the higher radial bound state solutions of $$ \Delta u +f(u)=0,\quad x\in \RR^n. \leqno(P) $$ We assume that the nonlinearity $f\in C(-\infty,\infty)$ is an odd function satisfying some convexity and growth…

We consider the uniqueness of the following positive solutions of $m$-Laplacian equation: \begin{equation} \left\{ \begin{aligned} -\Delta _m u&=\lambda u^{m-1}+u^{p-1} \quad \text{in} \quad \Omega\\ u&=0 \quad \text{on} \quad \partial…

We consider stable solutions to the equation $ -\Delta_p u =f(u) $ in a smooth bounded domain $\Omega\subset\mathbb{R}^n $ for a $ C^1 $ nonlinearity $f$. Either in the radial case, or for some model nonlinearities $f$ in a general domain,…

In this paper, we prove the existence of an infinite number of radial solutions of the $p$-$Laplacian$ equation $\Delta_p u + K(|x|) f(u) =0$ on the exterior of the ball of radius $R>0$ in ${\mathbb R}^{N}$ such that $u(|x|)\to 0$ as…

Assuming $B_{R}$ is a ball in $\mathbb R^{N}$, we analyze the positive solutions of the problem \[ \begin{cases} -\Delta u+u= |u|^{p-2}u, &\text{ in } B_{R},\newline \partial_{\nu}u=0,&\text{ on } \partial B_{R}, \end{cases} \] that branch…

We study the regularity of stable solutions to the problem $$ \left\{ \begin{array}{rcll} (-\Delta)^s u &=& f(u) & \text{in} \quad B_1\,, u &\equiv&0 & \text{in} \quad \mathbb R^n\setminus B_1\,, \end{array} \right. $$ where $s\in(0,1)$.…

We establish the uniqueness of positive radial solutions of $$\begin{cases} \Delta u +f(u)=0,\quad x\in A \\ u(x) =0 \quad x\in \partial A \end{cases} $$ where $A:=A_{a,b}=\{ x\in {\mathbb R}^n : a<|x|<b \}$, $0<a<b\le\infty$. We assume…

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 prove that the equation \begin{eqnarray*} -\Delta_p u =\lambda\Big( \frac{1} {u^\delta} + u^q + f(u)\Big)\;\text{ in } \, B_R(0) u =0 \,\text{ on} \; \partial B_R(0), \quad u>0 \text{ in } \, B_R(0) \end{eqnarray*} admits a weak radially…

We consider the problem of multiplicity and uniqueness of radial solutions of a nonlinear elliptic equation of the form \begin{eqnarray*} \begin{gathered} \Delta u +f(u)=0,\quad x\in \mathbb{R}^N, N\geq 2, \lim\limits_{|x|\to\infty}u(x)=0.…

Let $n\ge 3$, $0<m<\frac{n-2}{n}$, $\rho_1>0$, $\eta>0$, $\beta>\frac{m\rho_1}{n-2-nm}$, $\alpha=\alpha_m=\frac{2\beta+\rho_1}{1-m}$, $\beta_0>0$ and $\alpha_0=2\beta_0+1$. We use fixed point argument to give a new proof for the existence…

In a recent article by the authors [15] it was shown that wide classes of semilinear elliptic equations with exponential type nonlinearities admit singular radial solutions $U$ on the punctured disc in $\mathbb R^2$ which are also…

We consider the class of stable solutions to semilinear equations $-\Delta u=f(u)$ in a bounded smooth domain of $\mathbb{R}^n$. Since 2010 an interior a priori $L^\infty$ bound for stable solutions is known to hold in dimensions $n \leq 4$…

We consider the Gelfand problem in a bounded smooth domain $\Omega\subset \mathbb{R}^N$ with the Dirichlet boundary condition. We are interested in the boundedness of the extremal solution $u^*$. When the dimension $N\ge10$, it is known…

For $1<p<\infty$, we consider the following problem $$ -\Delta_p u=f(u),\quad u>0\text{ in }\Omega,\quad\partial_\nu u=0\text{ on }\partial\Omega, $$ where $\Omega\subset\mathbb R^N$ is either a ball or an annulus. The nonlinearity $f$ is…

We study the problem $-\Delta u=\lambda u-u^{-1}$ with a Neumann boundary condition; the peculiarity being the presence of the singular term $-u^{-1}$. We point out that the minus sign in front of the negative power of $u$ is particularly…