Existence and concentration of normalized solutions for $p$-Laplacian equations with logarithmic nonlinearity

Abstract

We investigate the existence and concentration of normalized solutions for a $p$-Laplacian problem with logarithmic nonlinearity of type \[ \left\{ \begin{array}{ll} \displaystyle -\varepsilon^pΔ_p u+V(x)|u|^{p-2}u=λ|u|^{p-2}u+|u|^{p-2}u\log|u|^p ~\text{in}~\mathbb R^N,\newline \displaystyle \int_{\mathbb R^N}|u|^pdx=a^p\varepsilon^N, \end{array} \right. \] where $a,\epsilon >0$, $\lambda \in \mathbb{R}$ is known as the Lagrange multiplier, ${\Delta }_p\cdot =\text{div}(|\nabla \cdot {|}^{p-2}\nabla \cdot )$ denotes the usual $p$-Laplacian operator with $2\le p<N$ and $V\in {\mathcal{C}}^0({\mathbb{R}}^N)$ 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 $\epsilon \to 0^{+}$ using variational method. Moreover, we also get the existence of normalized solutions for some logarithmic $p$-Laplacian equations involving mass-supercritical nonlinearities.

35 pages