Picone-type inequalities for nonlinear elliptic equations and their applications

Jaroš, Jaroslav; Kusano, Takaŝi; Yoshida, Norio

doi:10.1155/s1025583401000236

Cited by 9 publications

(14 citation statements)

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“…We note that Picone-type identity for the case A(x) = a(x)I was established in [22]. Problem 6.4 was studied in [31] under the assumption p = 2, and stronger statements were proved.…”

Section: Open Problemsmentioning

confidence: 94%

Ground state alternative for p-Laplacian with potential term

2006

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Let Ω be a domain in R d , d ≥ 2, and 1 < p < ∞. Fix V ∈ L ∞ loc (Ω). Consider the functional Q and its Gâteaux derivative Q ′ given byIn the latter case, v is (up to a multiplicative constant) the unique positive supersolution of the equation Q ′ (u) = 0 in Ω, and one has for Q an inequality of Poincaré type: there exists a positive continuous function W such that for every ψ ∈ C ∞ 0 (Ω) satisfying ψv dx = 0 there exists a constant C > 0 such that C −1 W |u| p dx ≤ Q(u) + C uψ dx p for all u ∈ C ∞ 0 (Ω). As a consequence, we prove positivity properties for the quasilinear operator Q ′ that are known to hold for general subcritical resp. critical second-order linear elliptic operators.

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“…We note that Picone-type identity for the case A(x) = a(x)I was established in [22]. Problem 6.4 was studied in [31] under the assumption p = 2, and stronger statements were proved.…”

Section: Open Problemsmentioning

confidence: 94%

Ground state alternative for p-Laplacian with potential term

2006

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“…. , m), we obtain the inequality (14) in [17] and (2.21) in [20]. Therefore, Theorem 3.1 is a partial extension of the theorems that are cited above.…”

Section: Remark 32 If We Omit the Damping Term That Ismentioning

confidence: 51%

“…There are many papers (and books) dealing with Picone identities and Picone-type inequalities for certain type differential equations. For example see [5,11,12,16,18,19,23,[28][29][30][35][36][37].…”

Section: Introductionmentioning

confidence: 99%

“…We refer to Kreith [21,22], Swanson [29] for Picone identities and Sturmian comparison theorems for linear elliptic equations, and to Allegretto [3], Allegretto and Huang [5,6], Bognár and Došlý [9], Dunninger [13], Jaroš et al [16,17,19,20], Kusano et al [23], Yoshida [36,38], Tiryaki [31], Tiryaki and Sahiner [33] for Picone identities, Sturmian comparison and/or oscillation theorems for half linear elliptic equations.…”

Section: Introductionmentioning

confidence: 99%

“…Thus, comparing the behavior of solutions of nonlinear equations with linear equations would help us to learn more about nonlinear equations. For instance in [17] with the nonlinear elliptic operators…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Sturm comparison theorems for some elliptic type equations via Picone-tpye inequalities

Tı̇ryakı̇¹,

Sahiner²,

Mısırlı³

2016

Proceedings of the 10'th Colloquium on the Qualitative Theory of Differential Equations (July 1–4, 2015, Szeged, Hungary) Edite

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Abstract. Sturm theorems have appeared as one of the fundamental subjects in qualitative theory to determine properties of the solutions of differential equations. Motivated by some recent developments for half-linear type elliptic equations, we obtain Piconetype inequalities for two pairs of elliptic type equations with damping and external terms in order to establish Sturmian comparison theorems. Some oscillation results are given as applications.

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Annulus criteria for mixed nonlinear elliptic differential equations

Şahiner

Zafer

2011

Mathematical and Computer Modelling

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a b s t r a c tNew oscillation criteria are obtained for forced second order elliptic partial differential equations with damping and mixed nonlinearities of the formIt is assumed that A = (a ij ) N×N is a real symmetric positive definite matrix function, b = (b i ) N×1 is a real vector function, a ij ∈ C 1+µ loc (Ω, R), and b i , c, c 1 , c 2 , e ∈ C µ loc (Ω, R) for all i, j, for some µ ∈ (0, 1).Examples are given to illustrate the results.

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Picone-type inequalities for nonlinear elliptic equations and their applications

Cited by 9 publications

References 0 publications

Ground state alternative for p-Laplacian with potential term

Ground state alternative for p-Laplacian with potential term

Sturm comparison theorems for some elliptic type equations via Picone-tpye inequalities

Annulus criteria for mixed nonlinear elliptic differential equations

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