2021/07/01 by Claudianor O. Alves, Alves, Claudianor O., Anass Ourraoui +3
Mathematics · Computer Science · #Nonlinear Partial Differential Equations #Advanced Mathematical Modeling in Engineering #Nonlinear Differential Equations Analysis
paper · pdf · doi:10.48550/arxiv.2107.00374
The aim of this paper is to establish two results about multiplicity of\nsolutions to problems involving the 1-Laplacian operator, with nonlinearities\nwith critical growth. To be more specific, we study the following problem \n
left
n -
Delta1 u +
xi
fracu|u| =
lambda |u|q-2u+|u|1^*-2u,\n
quad
textin
Omega,\n u=0,
quad
texton
partial
Omega.\n \
right. where \Ω is a smooth bounded domain in\n\ℝN, N \≥ 2 and \ξ \∈ 0,1 . Moreover, \λ > 0, q\n\∈ (1,1^*) and 1^*=\(N)/(N-1). The first main result establishes the\nexistence of many rotationally non-equivalent and nonradial solutions by\nassuming that \ξ=1, \Ω = x \∈ \ℝN ,: ,r < |x| < r+1 ,\nN\≥ 2, N not = 3 and r > 0. In the second one, \Ω is a smooth\nbounded domain, \ξ=0, and the multiplicity of solutions is proved through an\nabstract result which involves genus theory for functionals which are sum of a\nC1 functional with a convex lower semicontinuous functional.\n