The Maximum Principle

Pucci, Patrizia; Serrin, James

doi:10.1007/978-3-7643-8145-5

Cited by 545 publications

(562 citation statements)

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“…Referring to [30] for the case of the p-Laplace operator, and to [25] for the case of a broad class of quasilinear elliptic operators, we recall the following:…”

Section: Non-negative and Non-decreasing Function Such Thatmentioning

confidence: 99%

Monotonicity and one-dimensional symmetry for solutions of −Δ p u = f(u) in half-spaces

2011

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We prove a weak comparison principle in narrow domains for sub-super solutions to − p u = f (u) in the case 1 < p ≤ 2 and f locally Lipschitz continuous. We exploit it to get the monotonicity of positive solutions to − p u = f (u) in half spaces, in the case 2N +2 N +2 < p ≤ 2 and f positive. Also we use the monotonicity result to deduce some Liouville-type theorems. We then consider a class of sign-changing nonlinearities and prove a monotonicity and a one-dimensional symmetry result, via the same techniques and some general a-priori estimates. Mathematics Subject Classification (2000)

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“…Referring to [30] for the case of the p-Laplace operator, and to [25] for the case of a broad class of quasilinear elliptic operators, we recall the following:…”

Section: Non-negative and Non-decreasing Function Such Thatmentioning

confidence: 99%

Monotonicity and one-dimensional symmetry for solutions of −Δ p u = f(u) in half-spaces

2011

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“…Invoking Theorem 2.5.1, p. 34, of Pucci and Serrin [30] (see also Damascelli [10], p. 495) from (3.8) we infer that u 0 (z) < C + for all z ∈ Ω. Now suppose 2 < p < ∞.…”

Section: So We Havementioning

confidence: 99%

“…Also, if z ∈ K, then from (3.9) and the Harnack inequality of Pucci and Serrin [30], p. 154), we know that there exists ρ > 0 such that…”

Section: So We Havementioning

confidence: 99%

On p-superlinear equations with a nonhomogeneous differential operator

Aizicovici

Papageorgiou

Staicu

2012

Nonlinear Differ. Equ. Appl.

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“…From the strong maximum principle of Pucci-Serrin [17] (p. 111), we have u 0 (t) > 0 for all t ∈ (0, b). Invoking the boundary point theorem of PucciSerrin [17] (p. 120), we conclude that…”

Section: Remark 31mentioning

confidence: 99%

“…We have (a(|ũ (t)|)ũ (t)) ≤ c 2 ũ r−p ∞ũ (t) p a.e. in T, so, by using the results of Pucci-Serrin [17],ũ ∈ intĈ + . Next we show that in factũ ∈ intĈ + is the unique positive solution of (18).…”

Section: Proposition 32 If Hypotheses H(a) Hold Then Problemmentioning

confidence: 99%

Nonlinear nonhomogeneous periodic problems

Barletta

D’Aguì

Papageorgiou³

2016

Nonlinear Differ. Equ. Appl.

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We consider a nonlinear periodic problem driven by a nonhomogeneous differential operator and a Carathéodory reaction. We show that it has at least three solutions, two of constant sign and the third nodal. In the particular case of the scalar p−Laplacian and with a parametric reaction of equidiffusive type, we show that three solutions with precise sign exist if the parameter λ > λ1(p) = the first nonzero eigenvalue of the periodic scalar Laplacian. Finally, in the semilinear case (p = 2), we show that there is a second nodal solution, for a total of four nontrivial solutions all with sign information. Mathematics Subject Classification. 34B15, 34B18, 34C25, 58E05.

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The Maximum Principle

Cited by 545 publications

References 0 publications

Monotonicity and one-dimensional symmetry for solutions of −Δ p u = f(u) in half-spaces

Monotonicity and one-dimensional symmetry for solutions of −Δ p u = f(u) in half-spaces

On p-superlinear equations with a nonhomogeneous differential operator

Nonlinear nonhomogeneous periodic problems

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