修正非线性Schrdinger方程的变号解

Sign-changing solutions for modified nonlinear Schrodinger equation

本文考虑下列修正非线性Schrodinger方程：{△u＋1/2△u2-V（x）＋λ｜u｜p＋2 u=0 x∈R N u（x）→0, x→∞此方程出现在数学物理的一些模型中，数学上也有很多讨论．形式上方程有变分结构，但是很难找到合适的工作空间使得相应的泛函同时具有光滑性和满足紧性条件．本文引入p-Laplace项扰动，扰动问题的解作为原问题的近似解，并得到适当的估计，从而转向极限后得到原问题的解．本文证明方程有无穷多变号解满足积分约束∫R N｜u｜p dx=1,这时方程中常数λ作为Lagrange乘子出现．

In this paper, we consider the following modified nonlinear Schrodinger equation： {△u＋1/2△u2-V（x）＋λ｜u｜p＋2 u=0 x∈R N u（x）→0, x→∞ This type of equations have been involved in models of mathematical physics, and mathematical work has been done extensively in recent years. The problem has a formal variational structure, but it is difficult to find a suitable space in which the variational functional possesses both smoothness and compactness properties. We introduce a p-Laplacian perturbation approach, obtain solutions of the perturbed problems as approximate solutions of the original problem, and original problem. ∫R N｜u｜p dx=1, where establish suitable estimates for these solutions. Then we pass limits to get solutions of the prove the existence of infinitely many sign-changing solutions with the integral constraint λ appears as a Lagrangian multiplier