Browsing by Subject "normalized solutions"
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Item type:Article, Access status: Open Access , A comprehensive review on the existence of normalized solutions for four classes of nonlinear elliptic equations(Wydawnictwa AGH, 2025) Chen, Sitong; Tang, XianhuaThis paper provides a comprehensive review of recent results concerning the existence of normalized solutions for four classes of nonlinear elliptic equations: Schrödinger equations, Schrödinger-Poisson equations, Kirchhoff equations, and Choquard equationsItem type:Article, Access status: Open Access , Normalized solutions for critical Schrödinger equations involving (2,q)-Laplacian(Wydawnictwa AGH, 2025) Wei, Lulu; Song, YueqiangIn this paper, we consider the following critical Schrödinger equation involving $(2,q)$-Laplacian: \[\begin{cases} -\Delta u-\Delta_{q} u=\lambda u+\mu |u|^{\gamma-2}u+|u|^{2^*-2}u \quad\text{in }\mathbb{R}^N, \\ \int_{\mathbb{R}^N} |u|^{2}dx=a^2,\end{cases}\] where $\Delta_q u =\operatorname{div} (|\nabla u|^{q-2}\nabla u)$ is the $q$-Laplacian operator, $\mu, a\gt 0,$ $\lambda\in\mathbb{R}$, $\gamma\in(2,2^*)$, $q\in(\frac{2N}{N+2},2)$ and $N\geq3$. The meaningful and interesting phenomenon is the simultaneous occurrence of $(2,q)$-Laplacian and critical nonlinearity in the above equation. In order to obtain existence of multiple normalized solutions for such equation, we need to make a detailed estimate. More precisely, for the $L^2$-subcritical case, we use the truncation technique, concentration-compactness principle and the genus theory to get the existence of multiple normalized solutions. For the $L^2$-supercritical case, we obtain a couple of normalized solution for the above equation by a fiber map and concentration-compactness principle." /> <meta name="keywords" content="Schrödinger equation, $(2,q)$-Laplacian, variational methods, critical growth, normalized solutions