Related papers: Nonlinear curl-curl problems in $\mathbb{R}^3$

We look for ground states and bound states $E:\mathbb{R}^3\to\mathbb{R}^3$ to the curl-curl problem $$\nabla\times(\nabla\times E)= f(x,E) \qquad\hbox{in } \mathbb{R}^3$$ which originates from nonlinear Maxwell equations. The energy…

We consider the nonlinear curl-curl problem $\nabla\times\nabla\times U + V(x) U=f(x,|U|^2)U$ in $\mathbb{R}^3$ related to the nonlinear Maxwell equations with Kerr-type nonlinear material laws. We prove the existence of a symmetric…

We consider the nonlinear curl-curl problem $\nabla\times\nabla\times U + V(x) U= \Gamma(x)|U|^{p-1}U$ in $\mathbb{R}^3$ related to the nonlinear Maxwell equations for monochromatic fields. We search for solutions as minimizers (ground…

We look for multiple solutions $\mathbf{U}\colon\mathbb{R}^3\to\mathbb{R}^3$ to the curl-curl problem \[ \nabla\times\nabla\times\mathbf{U}=h(x,\mathbf{U}),\qquad x\in\mathbb{R}^3, \] with a nonlinear function…

We show the existence of the so-called semiclassical states $\mathbf{U}:\mathbb{R}^3\to\mathbb{R}^3$ to the following curl-curl problem $$ \varepsilon^2\; \nabla \times (\nabla \times \mathbf{U}) + V(x) \mathbf{U} = g(\mathbf{U}), $$ for…

We find solutions $E:\Omega\to\mathbb{R}^3$ of the problem \[ \left\{\begin{aligned} &\nabla\times(\nabla\times E) + \lambda E = \partial_E F(x,E) &&\quad \text{in}\Omega\\ &\nu\times E = 0 &&\quad \text{on}\partial\Omega \end{aligned}…

We investigate the existence of solutions $E:\mathbb{R}^3\to\mathbb{R}^3$ of the time-harmonic semilinear Maxwell equation $$\nabla\times(\nabla\times E) + V(x) E = \partial_E F(x,E) \quad \text{in}\mathbb{R}^3,$$ where…

We study the following nonlinear critical curl-curl equation \begin{equation}\label{eq0.1}\nabla\times \nabla\times U +V(x)U=|U|^{p-2}U+ |U|^4U,\quad x\in \mathbb{R}^3,\end{equation} where $V(x)=V(r, x_3)$ with $r=\sqrt{x_1^2+x_2^2}$ is…

We find solutions $E:\Omega\to\mathbb{R}^3$ of the problem \begin{eqnarray*} \left\{ \begin{aligned} &\nabla\times(\mu(x)^{-1}\nabla\times E) - \omega^2\epsilon(x) E = \partial_E F(x,E) &&\quad \text{in }\Omega\\%\newline &\nu\times E = 0…

The search for time-harmonic solutions of nonlinear Maxwell equations in the absence of charges and currents leads to the elliptic equation $$\nabla\times\left(\mu(x)^{-1} \nabla\times u\right) - \omega^2\varepsilon(x)u = f(x,u)$$ for the…

We are interested in the existence of normalized solutions to the problem \begin{equation*} \begin{cases} (-\Delta)^m u+\frac{\mu}{|y|^{2m}}u + \lambda u = g(u), \quad x = (y,z) \in \mathbb{R}^K \times \mathbb{R}^{N-K}, \\…

In this paper, we study the following $p(x)$-curl systems: \begin{eqnarray*} \begin{cases} \nabla\times(|\nabla\times \mathbf{u}|^{p(x)-2}\nabla\times \mathbf{u})+a(x)|\mathbf{u}|^{p(x)-2}\mathbf{u}=\lambda f(x,\mathbf{u})+\mu…

In this paper, using variational methods, we look for non-trivial solutions for the following problem $$ \begin{cases} -{\rm div}\left(a(|\nabla u|^2)\nabla u\right)=g(u), & \hbox{in }\mathbb{R}^N,\; N\geq 3, \\[1mm] u(x)\to 0, &\hbox{as…

In this paper, we study the existence of solutions for a critical time-harmonic Maxwell equation in nonlocal media. By introducing some suitable Coulomb spaces involving curl operator, we are able to obtain the ground state solutions of the…

We study the Schr\"odinger equations $-\Delta u + V(x)u = f(x,u)$ in $\mathbb{R}^N$ and $-\Delta u - \lambda u = f(x,u)$ in a bounded domain $\Omega\subset\mathbb{R}^N$. We assume that $f$ is superlinear but of subcritical growth and…

We look for solutions $E:\Omega\to\mathbb{R}^3$ of the problem $$ \left\{ \begin{aligned} &\nabla\times(\nabla\times E) +\lambda E = |E|^{p-2}E &&\quad \text{in }\Omega &\nu\times E = 0 &&\quad \text{on }\partial\Omega \end{aligned} \right.…

In this paper, we consider the following Kirchhoff type equation $$ -\left(a+ b\int_{\R^3}|\nabla u|^2\right)\triangle {u}+V(x)u=f(u),\,\,x\in\R^3, $$ where $a,b>0$ and $f\in C(\R,\R)$, and the potential $V\in C^1(\R^3,\R)$ is positive,…

Nonlinear Fokker-Planck equations endowed with curl drift forces are investigated. The conditions under which these evolution equations admit stationary solutions, which are $q$-exponentials of an appropriate potential function, are…

In this work we study an existence and multiplicity result for the following prescribed mean-curvature problem with critical growth $$ \left\{\begin{array}{rl} -\mbox{div}\biggl(\frac{\nabla u}{\sqrt{1+|\nabla u|^{2}}}\biggl) = \lambda…

In this paper, we apply the method of the Nehari manifold to study the Kirchhoff type equation \begin{equation*} -\Big(a+b\int_\Omega|\nabla u|^2dx\Big)\Delta u=f(x,u) \end{equation*} submitted to Dirichlet boundary conditions. Under a…