TY  - JOUR
ID  - Wei2025
LB  - Wei2025
AU  - Wei, Lulu
AU  - Song, Yueqiang
TI  - Normalized solutions for critical Schrödinger equations involving (2,q)-Laplacian
ST  - Normalized solutions for critical Schrödinger equations involving (2,q)-Laplacian
JO  - Opuscula Math.
JA  - Opuscula Math.
JF  - Opuscula Mathematica
PY  - 2025
VL  - 45
IS  - 5
SP  - 685
EP  - 716
DO  - https://doi.org/10.7494/OpMath.2025.45.5.685
UR  - https://doi.org/10.7494/OpMath.2025.45.5.685
AB  - In 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.
KW  - Schrödinger equation
KW  - \((2,q)\)-Laplacian
KW  - variational methods
KW  - critical growth
KW  - normalized solutions
SN  - 1232-9274
ER  -