Normalized solution to the nonlinear p-Laplacian equation with an L² constrain: mass supercritical case

2022/11/02 by Tian, Yulu, Wang, Deng-Shan, Zhao, Liang
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2211.01129

Abstract

In this paper, we study the existence of ground state solutions to the following p-Laplacian equation in some dimension N≥3 with an L² constraint: \begincases -Δ_pu+\vert u\vert^p-2u=f(u)-μu in ℝ^N,
\Vert u\Vert²_L²(ℝ^N)=m,
u∈ W^1,p(ℝ^N)∩ L²(ℝ^N), \endcases where -Δ_pu=div( \vert∇ u\vert^p-2∇ u ), 2≤ p0, μ∈ℝ will appear as a Lagrange multiplier and the continuous nonlinearity f satisfies mass supercritical conditions. We mainly study the behavior of ground state energy E_m with m>0 changing within a certain range and aim at extending nonlinear scalar field equation when p=2 and reducing the constraint condition of nonlinearity f.

Related

• Normalized solutions for p-Laplacian equations with potential
• Solutions with prescribed mass for L²-supercritical NLS equations under Neumann boundary conditions
• Normalized ground states for NLS equations with mass critical nonlinearities
• Ground state and multiple normalized solutions of quasilinear Schrödinger equations in the L²-supercritical case and the Sobolev critical case
• Normalized solution for p-Laplacian equation in exterior domain