Non-linear Schrödinger equation with non-local regional diffusion
The paper focuses on an existence result for the nonlinear Schrödinger equation written as \(\epsilon ^{2\alpha }(-\Delta )_{\rho }^{\alpha }u+u=f(u) \) in \(\mathbb{R}^{n}\), \(n\geq 2\), under appropriate hypotheses on \(\rho \) and on \(f\) and on some concentration properties of these weak solutions. Here, \(\epsilon >0\), \(\alpha \in (0,1)\) and \((-\Delta )_{\rho }^{\alpha }\) is a variational version of the non-local regional Laplace operator, with range of scope associated to \(\rho \in C(\mathbb{R}^{n},\mathbb{R}^{+})\). The authors introduce the energy functional associated to this nonlinear Schrödinger equation \[ I_{\rho }(u)=\frac{1}{2}\int_{\mathbb{R} ^{n}}\int_{B(0,\rho (x))}\frac{|u(x)-u(z)|^{2}}{|x-z|^{n+2\alpha }} dzdx+\int_{\mathbb{R}^{n}}u(x)^{2}dx-\int_{\mathbb{R}^{n}}F(u(x))dx, \] where \( F(t)=\int_{0}^{t}f(\xi )d\xi \). A weak solution in \(H^{\alpha }(\mathbb{R} ^{n})\) to the nonlinear Schrödinger equation is defined as a critical point of the energy functional \(I_{\rho }\). The first main result of the paper proves the existence of a non-trivial weak solution \(u\geq 0\) to the above-indicated nonlinear Schrödinger equation. For the proof, the authors apply the mountain pass theorem to the energy functional \(I_{\rho }\) and they use a comparison argument already introduced by the author, \textit{A. Quaas} and \textit{J. Tan} in [Proc. R. Soc. Edinb., Sect. A, Math. 142, No. 6, 1237-1262 (2012; Zbl 1290.35308)]. They also use some properties of the operator \((-\Delta )_{\rho }^{\alpha }\). The second main result proves a concentration behavior of ground states to the above-indicated Schrödinger equation. The authors indeed prove that when \(\epsilon\) goes to 0, there exists a subsequence of non-negative solutions which concentrate around a global minimum point of some limit concentration function \(\mathcal{H}\). The authors precise the speed of concentration. For the proof, they first define the notion of weak solution in \(H_{\rho }^{\alpha }(\mathbb{R}^{n})\) by means of a variational formulation. They then prove that the concentration function is continuous and satisfies further properties. Finally, they use comparison principles.
scientific article; zbMATH DE number 6481160
scientific article