Hylomorphic solitons on lattices
The paper concerns the \textit{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.
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