Related papers: Real solutions to the nonlinear Helmholtz equation…

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 study a stationary scattering problem related to the nonlinear Helmholtz equation $-\Delta u - k^2 u = f(x,u) \ \ \text{in $\mathbb{R}^N$,}$ where $N \ge 3$ and $k>0$. For a given incident free wave $\varphi \in L^\infty(\mathbb{R}^N)$,…

We set up a dual variational framework to detect real standing wave solutions of the nonlinear Helmholtz equation $$ -\Delta u-k^2 u =Q(x)|u|^{p-2}u,\qquad u \in W^{2,p}(\mathbb{R}^N) $$ with $N\geq 3$, $\frac{2(N+1)}{(N-1)}<…

In the first part of this paper, the existence of infinitely many $L^p$-standing wave solutions for the nonlinear Helmholtz equation $$ -\Delta u -\lambda u=Q(x)|u|^{p-2}u\quad\text{ in }\mathbb{R}^N $$ is proven for $N\geq 2$ and…

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 prove the existence and asymptotic expansion of a large class of solutions to nonlinear Helmholtz equations of the form \begin{equation*} (\Delta - \lambda^2) u = N[u], \end{equation*} where $\Delta = -\sum_j \partial^2_j$ is the…

This paper studies for large frequency number $k>0$ the existence and multiplicity of solutions of the semilinear problem $$ -\Delta u -k^2 u=Q(x)|u|^{p-2}u\quad\text{ in }\mathbb{R}^N, \quad N\geq 2. $$ The exponent $p$ is subcritical and…

This paper studies the nonlinear fractional Helmholtz equation \begin{equation}\label{main} (-\Delta)^{s} u-k^{2} u=Q(x)|u|^{p-2}u, ~~\mathrm{in}~~\mathbb{R}^{N},~~N\geq3, \end{equation} where $\frac{N}{N+1}<s<\frac{N}{2}$,…

Using a dual variational approach we obtain nontrivial real-valued solutions of the critical nonlinear Helmholtz equation $$ - \Delta u - k^{2}u = Q(x)|u|^{2^{\ast} - 2}u, \quad u \in W^{2,2^{\ast}}(\mathbb{R}^{N}) $$ for $N\geq 4$, where…

We prove new existence results for a Nonlinear Helmholtz equation with sign-changing nonlinearity of the form $$ - \Delta u - k^{2}u = Q(x)|u|^{p-2}u, \quad u \in W^{2,p}(\mathbb{R}^{N}) $$ with $k>0,$ $N \geq 3$, $p \in…

We investigate the existence of solutions to the fractional nonlinear Schr\"{o}dinger equation $(-\Delta)^s u = f(u)$ with prescribed $L^2$-norm $\int_{\mathbb{R}^N} |u|^2 \, dx =m$ in the Sobolev space $H^s(\mathbb{R}^N)$. Under fairly…

This paper concerns the existence of a nontrivial solution for the following problem \begin{equation} \left\{\begin{aligned} -\Delta u + V(x)u & \in \partial_u F(x,u)\;\;\mbox{a.e. in}\;\;\mathbb{R}^{N},\nonumber u \in…

We prove new results on the existence of positive radial solutions of the elliptic equation $-\Delta u= \lambda h(|x|,u)$ in an annular domain in $\mathbb{R}^{N}, N\geq 2$. Existence of positive radial solutions are determined under the…

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…

Given any $\mu_1, \mu_2\in {\mathbb C}$ and $\alpha >0$, we prove the local existence of arbitrarily smooth solutions of the nonlinear Klein-Gordon equation $\partial_{ tt } u - \Delta u + \mu_1 u = \mu_2 |u|^\alpha u$ on ${\mathbb R}^N$,…

This paper considers a pair of coupled nonlinear Helmholtz equations \begin{align*} -\Delta u - \mu u = a(x) \left( |u|^\frac{p}{2} + b(x) |v|^\frac{p}{2} \right)|u|^{\frac{p}{2} - 2}u, \end{align*} \begin{align*} -\Delta v - \nu v = a(x)…

We consider the nonlinear damped Klein-Gordon equation \[ \partial_{tt}u+2\alpha\partial_{t}u-\Delta u+u-|u|^{p-1}u=0 \quad \text{on} \ \ [0,\infty)\times \mathbb{R}^N \] with $\alpha>0$, $2 \le N\le 5$ and energy subcritical exponents…

We prove the existence of infinitely many solitary waves for the nonlinear Klein-Gordon or Schr\"odinger equation $$ \Delta u-u+ u^3 =0 , $$ in ${\bf R}^2$, which have finite energy and whose maximal group of symmetry reduces to the…

We consider the nonlinear Klein-Gordon equation $\partial_t^2u(x,t)-\partial_x^2u(x,t)+\alpha u(x,t)=\pm|u(x,t)|^{p-1}u(x,t)$ on a periodic metric graph (necklace graph) for $p>1$ with Kirchhoff conditions at the vertices. Under suitable…

In this paper, we investigate the existence of positive solution for the following class of elliptic equation $$ - \epsilon^{2}\Delta u +V(x)u= f(u) \,\,\,\, \mbox{in} \,\,\, \mathbb{R}^{N}, $$ where $\epsilon >0$ is a positive parameter,…