Normalized solutions for critical Schrödinger equations involving (2,q)-Laplacian

Lulu Wei
Yueqiang Song

Abstract. 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.

Keywords: Schrödinger equation, \((2,q)\)-Laplacian, variational methods, critical growth, normalized solutions.

Mathematics Subject Classification: 35J20, 35R03, 46E35.

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