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Existence and concentration of normalized solutions for $p$-Laplacian equations with logarithmic nonlinearity (2403.09366v1)

Published 14 Mar 2024 in math.AP

Abstract: We investigate the existence and concentration of normalized solutions for a $p$-Laplacian problem with logarithmic nonlinearity of type [ \left{ \begin{array}{ll} \displaystyle -\varepsilonp\Delta_p u+V(x)|u|{p-2}u=\lambda |u|{p-2}u+|u|{p-2}u\log|u|p ~\text{in}~\mathbb RN,\newline \displaystyle \int_{\mathbb RN}|u|pdx=ap\varepsilonN, \end{array} \right. ] where $a,\varepsilon> 0$, $\lambda\in\mathbb R$ is known as the Lagrange multiplier, $\Delta_p\cdot =\text{div} (|\nabla \cdot|{p-2}\nabla \cdot)$ denotes the usual $p$-Laplacian operator with $2\leq p < N$ and $V \in \mathcal{C}0(\mathbb RN)$ is the potential which satisfies some suitable assumptions. We prove that the number of positive solutions depends on the profile of $V$ and each solution concentrates around its corresponding global minimum point of $V$ in the semiclassical limit when $\varepsilon\to0+$ using variational method. Moreover, we also get the existence of normalized solutions for some logarithmic $p$-Laplacian equations involving mass-supercritical nonlinearities.

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