2023/11/01 by Wanting Qi, Xingyong Zhang, Qi, Wanting +1
Engineering · Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Navier-Stokes equation solutions #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.2311.00283
openalex publication_date 2023/11/01 · openalex created_date 2023/11/03 · openalex updated_date 2026/07/28
We investigate the multiplicity of solutions for a quasilinear scalar field equation with a nonhomogeneous differential operator defined by Su:=-div\ϕ(\fracu2+|∇ u|22)∇ u\+ϕ (\fracu2+|∇ u|22)u, where ϕ:[0,+∞)→ℝ is a positive continuous function. This operator is introduced by C. A. Stuart [Milan J. Math. 79 (2011), 327-341] and depends on not only ∇ u but also u. This particular quasilinear term generally appears in the study of nonlinear optics model which describes the propagation of self-trapped beam in a cylindrical optical fiber made from a self-focusing dielectric material. When the reaction term is concave-convex nonlinearities, by using the Nehari manifold and doing a fine analysis associated on the fibering map, we obtain that the equation admits at least one positive energy solution and negative energy solution which is also the ground state solution of the equation. We overcome two main difficulties which are caused by the nonhomogeneity of the differential operator S: (i) the almost everywhere convergence of the gradient for the minimizing sequence \un\; (ii) seeking the reasonable restrictions about S.