Normalized solutions to a class of (2, q)-Laplacian equationsin the strongly sublinear regime

2024/06/12 by Ding, Rui, Ji, Chao, Pucci, Patrizia · 1 citation
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2406.07985

Abstract

In this paper, we consider the existence and multiplicity of normalized solutions for the following (2, q)-Laplacian equation \\beginaligned amp;-Δu-Δ_q u+λu=g(u), x ∈ ℝ^N, amp;∫_ℝ^Nu² d x=c², \endaligned. where 10 is a constant. The nonlinearity g:ℝ→ ℝ is continuous and the behaviour of g at the origin is allowed to be strongly sublinear, i.e., lim _{s → 0} g(s) / s=-∞, which includes the logarithmic nonlinearity g(s)= s log s². We consider a family of approximating problems that can be set in H¹(ℝ^N)∩ D^{1, q}(ℝ^N) and the corresponding least-energy solutions. Then, we prove that such a family of solutions converges to a least-energy solution to the original problem. Additionally, under certain assumptions about g that allow us to work in a suitable subspace of H¹(ℝ^N)∩ D^{1, q}(ℝ^N), we prove the existence of infinitely many solutions of the above (2, q)-Laplacian equation.

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