相关论文: Ground States of a Nonlinear Curl-Curl Problem in …

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 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 survey recent results concerning ground states and bound states $u\colon\mathbb{R}^3\to\mathbb{R}^3$ to the curl-curl problem $$\nabla\times(\nabla\times u)+V(x)u= f(x,u) \quad\hbox{ in } \mathbb{R}^3,$$ which originates from the…

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…

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 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 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 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}, \\…

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 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 prove that the minimizer in the N\'ed\'elec polynomial space of some degree p > 0 of a discrete minimization problem performs as well as the continuous minimizer in H(curl), up to a constant that is independent of the polynomial degree…

We study a Sobolev-type inequality involving the $p$-curl operator in $\mathbb{R}^3$. We prove the existence of a minimizer which yields a solution to the $p$-curl-curl equation in the critical case. The problem is motivated both by…

We consider the following time-independent nonlinear $L^2$-critical Schr\"{o}dinger equation \[ -\Delta u(x)+V(x)u(x)-a|x|^{-b}|u|^{1+\frac{4-2b}{N}}=\mu u(x)\,\ \hbox{in}\,\ \mathbb{R}^N, \] where $\mu\in\mathbb{R}$, $a>0$, $N\geq 1$,…

This paper considers ground states of mass subcritical rotational nonlinear Schr\"{o}dinger equation \begin{equation*} -\Delta u+V(x)u+i\Omega(x^\perp\cdot\nabla u)=\mu u+\rho^{p-1}|u|^{p-1}u \,\ \text{in} \,\ \mathbb{R}^2, \end{equation*}…

In the paper we show the existence of ground state solutions to the nonlinear Born-Infeld problem \[ \mathrm{div}\, \left( \frac{\nabla u}{\sqrt{1-|\nabla u|^2}} \right) + f(u) = 0, \quad x \in \mathbb{R}^N \] in the zero and positive mass…

In this paper we study the existence of ground state solution and concentration of maxima for a class of strongly indefinite problem like $$ \left\{\begin{array}{l} -\Delta u+V(x)u=A(\epsilon x)f(u) \quad \mbox{in} \quad \R^{N}, \\ u\in…

We consider the nonlinear fractional problem \begin{align*} (-\Delta)^{s} u + V(x) u = f(x,u) &\quad \hbox{in $\mathbb{R}^N$} \end{align*} We show that ground state solutions converge (along a subsequence) in $L^2_{\mathrm{loc}}…

We study the existence of solutions of the following nonlinear Schr\"odinger equation \begin{equation*} -\Delta u + \Big(V(x)-\frac{\mu}{|x|^2}\Big) u = f(x,u) \hbox{ for } x\in\mathbb{R}^N\setminus\{0\}, \end{equation*} where…

We show that ground state solutions to the nonlinear, fractional problem \begin{align*} \left\{ \begin{array}{ll} (-\Delta)^{s} u + V(x) u = f(x,u) &\quad \mathrm{in} \ \Omega, \newline u = 0 &\quad \mathrm{in} \ \mathbb{R}^N \setminus…

We prove the existence of a ground state for some variational problems in Hilbert spaces, following the approach of Berestycki and Lions. Next, we examine the problem of constructing ground state solutions…