اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدNormalized solutions for the Schr{o}dinger equation with combined Hartree type and power nonlinearities
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We investigate normalized solutions for the Schr{o}dinger equation with combined Hartree type and power nonlinearities, namely begin{equation*} left{ begin{array}{ll} -Delta u+lambda u=gamma (I_{alpha }ast leftvert urightvert ^{p})|u|^{p-2}u+mu |u|^{q-2}u & quad text{in}quad mathbb{R}^{N}, int_{mathbb{R}^{N}}|u|^{2}dx=c, & end{array}% right. end{equation*} where $Ngeq 2$ and $c>0$ is a given real number. Under different assumptions on $gamma ,mu ,p$ and $q$, we prove several nonexistence, existence and multiplicity results. In particular, we are more interested in the cases when the competing effect of Hartree type and power nonlinearities happens, i.e. $gamma mu <0,$ including the cases $gamma <0,mu >0$ and $% gamma >0,mu <0.$ Due to the different strength of two types of nonlinearities, we find some differences in results and in the geometry of the corresponding functionals between these two cases.
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