Classification of stable solutions for non-homogeneous higher-order elliptic PDEs

Harrabi, Abdellaziz; Rahal, Belgacem; Hamdani, Mohamed Karim

doi:10.1186/s13660-017-1352-9

Cited by 2 publications

(3 citation statements)

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“…Therefore, we shall pay special attention to the delicate case of unbounded solutions. In fact, since (1.4) is weaker than the similar integral estimates obtained in [6,8,12,14,23] for r ≤ 4, then to provide nonexistence result we have to adapt the Moser iteration technique which requires that u evolves less rapidly than an exponential growth:…”

Section: Resultsmentioning

confidence: 99%

“…There are few papers considering the nonhomogeneous nonlinearities (see [12] and the references therein). Particularly, in an elegant paper, Dupaigne-Farina investigated the nonhomogeneous convex nonlinearities for r = 1 [7].…”

Section: Introductionmentioning

confidence: 99%

“…We will establish various nonexistence results for stable solutions of (1.1) (respectively (1.2)) or solutions which are stable outside a compact set, for a large class of global superlinear nonlinearities. The proof of the integral estimate for r ≤ 3 is closely related to an iterative use of Young and Hölder's inequalities (see [6,8,11,12,14,23]), but these kinds of arguments seem to be difficultly extendable to the general higher-order operator. Our approach relies on a crucial idea, borrowed from [10,19], where an appropriate family of test functions and weighted seminorms are used to obtain some local L p -W 2r , p estimates for solutions of the linear polyharmonic problems.…”

Section: Introductionmentioning

confidence: 99%

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High-order Bahri–Lions Liouville-type theorems

Harrabi

2019

Annali di Matematica

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Consider the following polyharmonic equations: (−) r u = f (u) in O, (0.1) where O = R n or O = R n + with the Dirichlet boundary conditions, r ∈ N * and n ≥ 2r + 1. We prove some Liouville-type theorems classifying stable (or stable at infinity) solutions, possibly unbounded and sign-changing. Regarding the class of stable solutions, we focus on the case of superlinear nonlinearities f with subcritical or critical growth near zero, like f (s) = |s| p−1 s(1 + c 0 |s| q) or f (s) = |s| p−1 s exp(s 2), where 1 < p ≤ n + 2r n − 2r , q > 0 and c 0 ≥ 0. Our approach to get the main integral estimates makes use of delicate analysis with appropriate test functions and weighted seminorms. We also establish a variant of Pohozaev identity (Pohozaev in Sov Math Dokl 5:1408-1411, 1965). This permits us to get classification result for stable at infinity solutions under the global subcritical condition: 2n n − 2r F(s) − f (s)s > 0, ∀s = 0, where F(s) = s 0 f (t)dt. Our assumptions can be verified by many nonlinearities very close to the critical growth, like f (s) = c|s| p−1 s + |s| 4r n−2r s ln q (s 2 + a) , where 1 < p < n+2r n−2r , q ≥ 0, c ≥ 0 and a > 1, with 4r n−2r − 2q a ln(a) > 0; or c > 0 if q = 0.

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