Multiple solutions for a class of nonlinear Schrödinger equations
The authors of this very interesting paper consider the nonlinear Schrödinger equation having the form \[ -\triangle u+V(x)u=g(x,u) \] for \(x\in \mathbb R ^{N}\), where \(u(x)\to 0\) as \(| x| \to \infty \). This equation arises in numerous applications in physics and chemistry. It turns out that there exist infinitely many homoclinic type solutions under the following assumptions: (1) \(V\in C^1(\mathbb R ^{N},\mathbb R )\), \(V(x)>0\) and is 1-periodic in \(x_j\) \((j=1,\dots,N)\); (2) \(g\in C^2(\mathbb R ^{N}\times \mathbb R ,\mathbb R )\) and \(g(x,u)\) is 1-periodic in \(x_j\) \((j=1,\dots,N)\); (3) \(g(x,0)=0=g_u(x,0)\) and \(G(x,u)\equiv \int_{0}^{u}g(x,s)ds\), \(2^{*}\equiv 2N/(N-2)\) for \(N\geq 3\); (4) \(G(x,u)\geq 0\), \(g_u(x,u)\to V_{\infty }(x)\) as \(| u| \to \infty \) uniformly in \(x\), where \(V_{\infty }(x)\) is 1-periodic in \(x_j\) \((j=1,\dots,N)\) and inf \(V_{\infty }(\mathbb R ^{N})>\bar \lambda \); (5) \(\tilde G(x,u)\equiv (1/2)g(x,u)u-G(x,u)\geq 0\) and there is \(\rho\in (0,\underline{ \lambda })\) such that \(\tilde G(x,u)\geq \rho \) whenever \(g(x,u)/u\underline{ \lambda } -\rho \). Then there exist infinitely many \(\mathbb{Z}^N\)-distinct solutions. Here two solutions \(u_1,u_2\) of the equation under consideration are said to be geometrically distinct (or \(\mathbb{Z}^N\)-distinct) if \(u_1\neq \tau_ku_2\) for all \(k=(k_1,\dots,k_N)\in\mathbb{Z}^N\), where \((\tau_ku)(x)=u(x_1+k_1,\dots,x_N+k_N)\). The second result concerns the case under assumptions (1)-(3) and a modification of (4) and (5). Then the authors prove again that there exist infinitely many \(\mathbb{Z}^N\)-distinct solutions.
scientific article; zbMATH DE number 2137238
scientific article