# Standing waves in the Maxwell-Schrï¿½dinger equation and an optimal configuration problem

## Related Articles

- Standing Waves for Nonlinear SchrÃ¶dinger Equations with a General Nonlinearity. Byeon, Jaeyoung; Jeanjean, Louis // Archive for Rational Mechanics & Analysis;Aug2007, Vol. 185 Issue 2, p185
For elliptic equations Îµ2Î” u âˆ’ V( x) u + f( u) = 0, x âˆˆ R N , N â‰§ 3, we develop a new variational approach to construct localized positive solutions which concentrate at an isolated component of positive local minimum points of V, as Îµ â†’ 0, under conditions on...

- Thick clusters for the radially symmetric nonlinear SchrÃ¶dinger equation. Felmer, Patricio; Martínez, Salomé // Calculus of Variations & Partial Differential Equations;Feb2008, Vol. 31 Issue 2, p231
This article is devoted to the study of radially symmetric solutions to the nonlinear SchrÃ¶dinger equation where B is a ball in $${\mathbb{R}}^N$$ , 1 < p < ( N + 2)/( N âˆ’ 2), N â‰¥ 3 and the potential V is radially symmetric. We construct positive clustering solutions in an annulus...

- Exponentially accurate error estimates of quasiclassical eigenvalues. II. Several dimensions. Toloza, J. H. // Journal of Mathematical Physics;Jul2003, Vol. 44 Issue 7, p2806
We study the behavior of truncated Rayleigh-SchrÃ¶dinger series for low-lying eigenvalues of the time-independent SchrÃ¶dinger equation, in the semiclassical limit &hstroke;&nsarr; 0. In particular we prove that if the potential energy satisfies certain conditions, there is an optimal...

- Convergence of Logarithmic Quantum Mechanics to the Linear One. G�rka, Przemyslaw // Letters in Mathematical Physics;Sep2007, Vol. 81 Issue 3, p253
We study the nonlinear logarithmic Schrï¿½dinger equation in three dimensions. We establish the existence of the solutions of general quasi-linear Schrï¿½dinger equations. Finally, we show the convergence of the logarithmic quantum mechanics to the linear regime.

- A Counterexample to Dispersive Estimates for Schrï¿½dinger Operators in Higher Dimensions. Goldberg, Michael; Visan, Monica // Communications in Mathematical Physics;Aug2006, Vol. 266 Issue 1, p211
In dimension n > 3 we show the existence of a compactly supported potential in the differentiability class Ca, a < n-3/2, for which the solutions to the linear Schrï¿½dinger equation in Rn, -i...tu = - ?u + Vu, u(0) = f, fail to satisfy an evolution estimate of the form "Multiple line...

- Multi-peak positive solutions for nonlinear Schrï¿½dinger equations with critical frequency. Sato, Yohei // Calculus of Variations & Partial Differential Equations;Jul2007, Vol. 29 Issue 3, p365
We study the nonlinear Schrï¿½dinger equations: $$-\epsilon^{2}\Delta u + V(x)u=u^p,\quad u > 0\quad \mbox{in } {\bf R}^{N},\quad u\in H^{1} ({\bf R}^{N}).$$ where p > 1 is a subcritical exponent and V( x) is nonnegative potential function which has ï¿½critical frequencyï¿½...

- Bound states for semilinear Schrï¿½dinger equations with sign-changing potential. Ding, Yanheng; Szulkin, Andrzej // Calculus of Variations & Partial Differential Equations;Jul2007, Vol. 29 Issue 3, p397
We study the existence and the number of decaying solutions for the semilinear Schrï¿½dinger equations $${-\varepsilon^{2}\Delta u + V(x)u = g(x,u)}$$ , $${\varepsilon > 0}$$ small, and $${-\Delta u + \lambda V(x)u = g(x,u)}$$ , $${\lambda > 0}$$ large. The potential V may change sign and g...

- High Frequency Solutions for the Singularly-Perturbed One-Dimensional Nonlinear SchrÃ¶dinger Equation. Felmer, Patricio; Martínez, Salomé; Tanaka, Kazunaga; Rabinowitz, P. // Archive for Rational Mechanics & Analysis;Nov2006, Vol. 182 Issue 2, p333
This article is devoted to the nonlinear SchrÃ¶dinger equation [InlineMediaObject not available: see fulltext.] when the parameter Îµ approaches zero. All possible asymptotic behaviors of bounded solutions can be described by means of envelopes, or alternatively by adiabatic profiles. We...

- Energy Growth in Schrï¿½dinger's Equation with Markovian Forcing. Erdo>ilde;an, M. Burak; Killip, Rowan; Schlag, Wilhelm // Communications in Mathematical Physics;Sep2003, Vol. 240 Issue 1/2, p1
Schrï¿½dinger's equation is considered on a one-dimensional torus with time dependent potential v(?,t)=?V(?)X(t), where V(?) is an even trigonometric polynomial and X(t) is a stationary Markov process. It is shown that when the coupling constant ? is sufficiently small, the average kinetic...