Related papers: Solutions without any symmetry for semilinear elli…
For the one-dimensional nonlinear damped Klein-Gordon equation \[ \partial_{t}^{2}u+2\alpha\partial_{t}u-\partial_{x}^{2}u+u-|u|^{p-1}u=0 \quad \mbox{on $\mathbb{R}\times\mathbb{R}$,}\] with $\alpha>0$ and $p>2$, we prove that any global…
We consider the damped nonlinear Klein-Gordon equation: \begin{align*} \partial_{t}^2u-\Delta u+2\alpha \partial_{t}u+u-|u|^{p-1}u=0, \ & (t,x) \in \mathbb{R} \times \mathbb{R}^d, \end{align*} where $\alpha>0$, $1\leq d\leq 5$ and energy…
Existence results for radially symmetric oscillating solutions for a class of nonlinear autonomous Helmholtz equations are given and their exact asymptotic behavior at infinity is established. Some generalizations to nonautonomous radial…
We consider finite energy solutions to the nonlinear Schroedinger equation and nonlinear Klein--Gordon equation and find the condition on the nonlinearity so that the standard, one-frequency solitary waves are the only solutions with…
We study the existence of positive solutions to the quasilinear elliptic problem -\epsilon \Delta u+V(x)u-\epsilon k(\Del(|u|^{2}))u=g(u), \quad u>0, x \in R^N, where g has superlinear growth at infinity without any restriction from above…
In this paper we establish uniqueness criteria for positive radially symmetric finite energy solutions of semilinear elliptic systems of the form \begin{align*} \begin{aligned} - \Delta u &= f(|x|,u,v)\quad\text{in}\R^n, - \Delta v &=…
In this paper we establish existence and stability results concerning fully nontrivial solitary-wave solutions to 3-coupled nonlinear Schr\"odinger system \[ i\partial_t u_{j}+\partial_{xx}u_{j}+ \left(\sum_{k=1}^{3} a_{kj}…
In this paper, we study real solutions of the nonlinear Helmholtz equation $$ - \Delta u - k^2 u = f(x,u),\qquad x\in \R^N $$ satisfying the asymptotic conditions $$ u(x)=O(|x|^{\frac{1-N}{2}}) \quad \text{and} \quad \frac{\partial^2…
We consider the standing-wave problem for a nonlinear Schr\"{o}dinger equation, corresponding to the semilinear elliptic problem \begin{equation*} -\Delta u+V(x)u=|u|^{p-1}u,\ u\in H^1(\mathbb{R}^2), \end{equation*} where $V(x)$ is a…
$\newcommand\normt[1]{\left\lVert#1\right\rVert_{L^2}} \newcommand\normo[1]{\left\lVert#1\right\rVert_{H^1}} \newcommand\normpro[1]{\left\lVert#1\right\rVert_{E}}$ We consider the focusing nonlinear Klein-Gordon (NLKG) equation…
We describe completely 2-solitary waves related to the ground state of the nonlinear damped Klein-Gordon equation \begin{equation*} \partial_{tt}u+2\alpha\partial_{t}u-\Delta u+u-|u|^{p-1}u=0 \end{equation*} on $\bf R^N$, for $1\leq N\leq…
The long-time asymptotics is analyzed for all finite energy solutions to a model U(1)-invariant nonlinear Klein-Gordon equation in one dimension, with the nonlinearity concentrated at a point. Our main result is that each finite energy…
We prove the existence of positive solutions to a sys- tem of k non-linear elliptic equations corresponding to standing- wave k-uples solutions to a system of non-linear Klein-Gordon equations. Our solutions are characterised by a small…
We are interested in the nonlinear damped Klein-Gordon equation \[ \partial_t^2 u+2\alpha \partial_t u-\Delta u+u-|u|^{p-1}u=0 \] on $\mathbb{R}^d$ for $2\le d\le 5$ and energy sub-critical exponents $2 < p < \frac{d+2}{d-2}$. We construct…
We analyze the semilinear elliptic equation $\Delta u=\rho(x) f(u)$, $u>0$ in ${\mathbf R}^D$ $(D\ge3)$, with a particular emphasis put on the qualitative study of entire large solutions, that is, solutions $u$ such that…
We consider the following nonlinear Schrodinger equation [{l} \Delta u-(1+\delta V)u+f(u)=0 in \R^N, u>0 in \R^N, u\in H^1(\R^N).] where $V$ is a potential satisfying some decay condition and $ f(u)$ is a superlinear nonlinearity satisfying…
We consider the damped nonlinear Klein-Gordon equation with a delta potential \begin{align*} \partial_{t}^2u-\partial_{x}^2u+2\alpha \partial_{t}u+u-\gamma {\delta}_0u-|u|^{p-1}u=0, \ & (t,x) \in \mathbb{R} \times \mathbb{R}, \end{align*}…
In this paper, we study the existence of positive solutions to the nonlinear elliptic system, which is derived from taking the nonrelativistic limit of the nonlinear Maxwell-Klein-Gordon equations under the decomposition of waves functions…
In this paper we study the semilinear elliptic problem $$ -\Delta u -k^2u=Q|u|^{p-2}u\quad\text{ in }\mathbb{R}^2, $$ where $k>0$, $p\geq 6$ and $Q$ is a bounded function. We prove the existence of real-valued $W^{2,p}$-solutions, both for…
We consider the damped nonlinear Klein-Gordon equation: \begin{align*} \partial_{t}^2u-\Delta u+2\alpha \partial_{t}u+u-|u|^{p-1}u=0, \ & (t,x) \in \mathbb{R} \times \mathbb{R}^d, \end{align*} where $\alpha>0$, $2\leq d\leq 5$ and energy…