Sobolev versus Hölder local minimizers and existence of multiple solutions for a singular quasilinear equation

Giacomoni, Jacques; Schindler, Ian; Takáč, Peter

doi:10.2422/2036-2145.2007.1.07

Cited by 109 publications

(116 citation statements)

References 21 publications

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“…We start by showing that every weak solution of a general anisotropic singular problem is essentially bounded. This property extends to the anisotropic singular framework results obtained by Fan and Zhao [16], Giacomoni, Schindler and Takač [25], Byun and Ko [11]. So, we consider the following anisotropic Dirichlet problem…”

Section: 1mentioning

confidence: 61%

“…For isotropic equations, such problems were investigated by Ghergu and Rȃdulescu [23,24], Haitao [27], Sun, Wu and Long [55] (semilinear equations driven by the Laplacian) and by Giacomoni, Schindler and Takač [25], Papageorgiou, Rȃdulescu and Repovš [40], Papageorgiou, Vetro and Vetro [45], Papageorgiou and Winkert [49]. Very recently, Papageorgiou, Rȃdulescu and Repovš [41] studied isotropic singular problems driven by a more general not necessary homogeneous, differential operator.…”

Section: Introductionmentioning

confidence: 99%

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Anisotropic singular double phase Dirichlet problems

Papageorgiou¹,

Rădulescu²,

Zhang³

2021

DCDS-S

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<p style='text-indent:20px;'>We consider an anisotropic double phase problem with a reaction in which we have the competing effects of a parametric singular term and a superlinear perturbation. We prove a bifurcation-type result describing the changes in the set of positive solutions as the parameter varies on <inline-formula><tex-math id="M1">\begin{document}$ \mathring{\mathbb{R}}_+ = (0, +\infty) $\end{document}</tex-math></inline-formula>. Our approach uses variational tools together with truncation and comparison techniques as well as several general results of independent interest about anisotropic equations, which are proved in the Appendix.</p>

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Section: 1mentioning

confidence: 61%

Section: Introductionmentioning

confidence: 99%

Anisotropic singular double phase Dirichlet problems

Papageorgiou¹,

Rădulescu²,

Zhang³

2021

DCDS-S

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“…Since f ≥ 0 by hypothesis H 1 (i), we have that u λ ≤ u and because u ∈ int C 1 0 ( ) + , there exists c 6 > 0 such that c 6d ≤ u, see Papageorgiou-Rȃdulescu-Repovš [17, p. 274]. This fact, hypothesis H 1 (i) and Theorem B1 of Saoudi-Ghanmi [26] (see also Giacomoni-Schindler-Takáč [10]), we have that u ∈ int C 1 0 ( ) + . Therefore,…”

Section: Positive Solutionsmentioning

confidence: 91%

Positive Solutions for Singular Anisotropic (p, q)-Equations

Papageorgiou

Winkert

2021

J Geom Anal

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In this paper, we consider a Dirichlet problem driven by an anisotropic (p, q)-differential operator and a parametric reaction having the competing effects of a singular term and of a superlinear perturbation. We prove a bifurcation-type theorem describing the changes in the set of positive solutions as the parameter moves. Moreover, we prove the existence of a minimal positive solution and determine the monotonicity and continuity properties of the minimal solution map.

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“…See Giacomoni-Schindler-Takáč [31], Bal-Garain [6] and the references therein. Anisotropic p-Laplace equation has been a topic of considerable attention in the recent years.…”

Section: Introductionmentioning

confidence: 99%