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

2024/03/14 by Liejun Shen, Marco Squassina, Shen, Liejun +1
Mathematics · Computer Science · #Nonlinear Differential Equations Analysis #Advanced Mathematical Modeling in Engineering #Differential Equations and Boundary Problems

paper · pdf · doi:10.48550/arxiv.2403.09366

Abstract

We investigate the existence and concentration of normalized solutions for a p-Laplacian problem with logarithmic nonlinearity of type \ -ε^pΔ_p u+V(x)|u|^p-2u=λ|u|^p-2u+|u|^p-2ulog|u|^p ~in~\mathbb R^N,\newline ∫_{\mathbb R^N}|u|^pdx=a^pε^N, . where a,ε> 0, λ∈\mathbb R is known as the Lagrange multiplier, Δ_p⋅ =div (|∇ ⋅|^p-2∇ ⋅) denotes the usual p-Laplacian operator with 2≤ p < N and V ∈ C⁰(\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 ε→0⁺ 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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