Existence of solution of some p, q${p, q}$‐Laplacian system under local superlinear conditions. Issue 1 (7th January 2022)

• Record Type: Journal Article
• Authors: Cerda, Patricio; Souto, Marco; Ubilla, Pedro
• Is Part Of: Mathematische Nachrichten. Volume 295:Issue 1(2022)
• Page Start: 44
• Page End: 57
• Publication Date: 2022-01-07
• Subjects: a priori bounds -- fixed point index -- positive radial solutions -- quasi‐linear elliptic systems; Mathematics -- Periodicals; 510.5
• DOI: 10.1002/mana.201900424
• Languages: English
• ISSNs: 0025-584X

Abstract

Abstract: We study existence of positive radial solutions for the following class of quasi‐linear elliptic systems − Δ p u = f ( | x |, u, v ) in B, − Δ q v = g ( | x |, u, v ) in B, ( u, v ) = ( 0, 0 ) on ∂ B, \begin{equation*}\hspace*{13.5pc} \left\lbrace \def\eqcellsep{&}\begin{array}{rcll} -\Delta _p u &=& f(|x|, u, v) & \text{in}\quad B, \\ -\Delta _q v &=& g(|x|, u, v) & \text{in}\quad B, \\ (u, v) &=& (0, 0) & \text{on}\quad \partial B, \end{array} \right. \end{equation*} where the nonlinearities f, g ∈ C ( B × [ 0, + ∞ ) 2 ; [ 0, + ∞ ) ) $ f, g \in C\big (B \times [0, +\infty )^2;\, [0, +\infty )\big )$ satisfy some local superlinear property at + ∞ $+\infty$ . Here B is the unity ball in R N ${\mathbb {R}}^N$ and 1 < p, q ≤ 2 $1< p, q\le 2$ . Some difficulties here are that this system, in general, is non‐variational and that p ≠ q $p \ne q$ might occur. Thus, our strategy to obtain a positive solution is through a priori bound methods. One of the important results in this article is that we obtain these estimates, assuming superlinearity hypotheses on the variables u and v at infinity just in some subset B o $B_o$ of the ball, allowing us assuming, for example, supercritical growths on u, v $u, v$ outside of that subset. We also note that we can consider nonlinearities which are not necessary asymptotic to a power at + ∞ $+\infty$ and so it is not possible to use the classic Liouville type nonexistence results.