## Hylomorphic solitons on lattices.(English)Zbl 1205.37081

The paper concerns the hylomorphic solitons whose existence is related to the ratio energy/charge and they include $$Q$$-balls, which are spherically symmetric solutions of the nonlinear Klein-Gordon equation, as well as solitary waves and vortices which occur, by the same mechanism, in the nonlinear Schrödinger equation and in gauge theories. A general existence theorem for this class of solitons is given. The abstract result is applied to the following nonlinear Schrödinger equation $i\frac{\partial \psi}{\partial t}=- \frac{1}{2}\Delta\psi + V(x)\psi + \frac{1}{2}W'(\psi),$ where $$\psi:\mathbb{R}^N\to \mathbb{C}$$ $$(N\geq 3)$$, $$V: \mathbb{R}^N\to \mathbb{R}$$ with the positivity and lattice invariance, $$W:\mathbb{C}\to \mathbb{R}$$ such that $$W(s)=F(|s|)$$ for some smooth function $$F:[0,\infty)\to \mathbb{R}$$, $W'(s)=\frac{\partial W}{\partial s_1}+i\frac{\partial W}{\partial s_2}, \quad s=s_1+is_2,\eqno (1)$ and $$W(s)=\frac{1}{2} (W''(0))^2s^2+N(s)$$ $$(N(s)=o(s^2))$$ with the positivity, nondegeneracy, hylomorphy, and growth condition. Moreover, the abstract result is applied to the following nonlinear Klein-Gordon equation $\square\psi + W'(x,\psi)=0,$ where $$\psi:\mathbb{R}^N\to \mathbb{C}$$ $$(N\geq 3)$$, $$W:\mathbb{R}^N\times \mathbb{C}\to \mathbb{R}$$, $$W'$$ is the derivative with respect to the second variable as in (1), and $$W(x,s)=\frac{1}{2} h(x)^2s^2+N(x,s)$$ ($$h(x)\in L^\infty$$, $$h(x)\geq h_0>0$$ and $$N(x,s)=o(s^2)$$) with the positivity, nondegeneracy, hylomorphy, and growth condition.

### MSC:

 37K40 Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems 35Q51 Soliton equations 35Q55 NLS equations (nonlinear Schrödinger equations) 35J50 Variational methods for elliptic systems 35J08 Green’s functions for elliptic equations 47J30 Variational methods involving nonlinear operators
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