We study a class of semilinear SchrSdinger equation with electromagnetic fields and the nonlinearity term involving critical growth. We assume that the potential of the equation includes a parameter A and can be negat...We study a class of semilinear SchrSdinger equation with electromagnetic fields and the nonlinearity term involving critical growth. We assume that the potential of the equation includes a parameter A and can be negative in some domain. Moreover, the potential behaves like potential well when the parameter A is large. Using variational methods combining Nehari methods, we prove that the equation has a least energy solution which, as the parameter A becomes large, localized near the bottom of the potential well. Our result is an extension of the corresponding result for the SchrSdinger equation which involves critical growth but does not involve electromagnetic fields.展开更多
We consider the quasilinear Schrdinger equations of the form-ε~2？u ＋ V（x）u- ε~2？（u2）u = g（u）, x ∈ R^N,where ε 〉 0 is a small parameter, the nonlinearity g（u） ∈ C^1（R） is an odd function with subcrit...We consider the quasilinear Schrdinger equations of the form-ε~2？u ＋ V（x）u- ε~2？（u2）u = g（u）, x ∈ R^N,where ε 〉 0 is a small parameter, the nonlinearity g（u） ∈ C^1（R） is an odd function with subcritical growth and V（x） is a positive Hlder continuous function which is bounded from below, away from zero, and infΛV（x） 0 such that for all ε∈（0, ε0],the above mentioned problem possesses a sign-changing solution uε which exhibits concentration profile around the local minimum point of V（x） as ε→ 0~＋.展开更多
基金supported by Fundamental Research Funds for the Central Universities and National Natural Science Foundation of China(Grant No.11171028)
文摘We study a class of semilinear SchrSdinger equation with electromagnetic fields and the nonlinearity term involving critical growth. We assume that the potential of the equation includes a parameter A and can be negative in some domain. Moreover, the potential behaves like potential well when the parameter A is large. Using variational methods combining Nehari methods, we prove that the equation has a least energy solution which, as the parameter A becomes large, localized near the bottom of the potential well. Our result is an extension of the corresponding result for the SchrSdinger equation which involves critical growth but does not involve electromagnetic fields.
基金supported by National Natural Science Foundation of China(Grant Nos.11371160 and 11328101)the Program for Changjiang Scholars and Innovative Research Team in University(Grant No.#IRT13066)
文摘We consider the quasilinear Schrdinger equations of the form-ε~2？u ＋ V（x）u- ε~2？（u2）u = g（u）, x ∈ R^N,where ε 〉 0 is a small parameter, the nonlinearity g（u） ∈ C^1（R） is an odd function with subcritical growth and V（x） is a positive Hlder continuous function which is bounded from below, away from zero, and infΛV（x） 0 such that for all ε∈（0, ε0],the above mentioned problem possesses a sign-changing solution uε which exhibits concentration profile around the local minimum point of V（x） as ε→ 0~＋.
