Positive maximal and minimal solutions for non-homogeneous elliptic equations depending on the gradient

Figueiredo, Giovany M.; Madeira, Gustavo Ferron

doi:10.1016/j.jde.2020.10.033

Cited by 23 publications

(7 citation statements)

References 56 publications

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“…The presence of the gradient in the nonlinear term, called convection term, makes variational methods not applicable. Among the techniques used to study problems with a convection term, we cite the following: topological degree method [2,20], theory of pseudomonotone operators [12], fixed point theorems [5,24], sub-and supersolution methods [10,11,13,18], approximation methods [23], or a combination of the techniques above [4,9,17]. We deal with existence, regularity, and sign of the solutions to (1.1).…”

Section: Introductionmentioning

confidence: 99%

Regular solutions for nonlinear elliptic equations, with convective terms, in Orlicz spaces

Barletta

Tornatore

2023

Mathematische Nachrichten

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We establish some existence and regularity results to the Dirichlet problem, for a class of quasilinear elliptic equations involving a partial differential operator, depending on the gradient of the solution. Our results are formulated in the Orlicz-Sobolev spaces and under general growth conditions on the convection term. The sub-and supersolutions method is a key tool in the proof of the existence results.

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Section: Introductionmentioning

confidence: 99%

Regular solutions for nonlinear elliptic equations, with convective terms, in Orlicz spaces

Barletta

Tornatore

2023

Mathematische Nachrichten

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“…For the nonlinear elliptic problems with gradient dependence we refer to the following papers: Averna-Motreanu-Tornatore [1], Bai [2], Bai-Gasiński-Papageorgiou [3], Faraci-Motreanu-Puglisi [4], Gasiński-Papageorgiou [5,6], Gasiński-Winkert [7], Motreanu-Motreanu-Moussaoui [8], Guarnotta-Marano-Motreanu [9], Papageorgiou-Rȃdulescu-Repovš [10], Faraci-Puglisi [11], Figueiredo-Madeira [12], Papageorgiou-Rǎdulescu-Repovš [13], Tanaka [14], Guarnotta-Marano [15], Liu-Motreanu-Zeng [16], Marano-Winkert [17], Araujo-Faria [18], Bai-Papageorgiou-Zeng [19]. None of the above papers deals with multivalued or obstacle problems.…”

Section: Introductionmentioning

confidence: 99%

Nonlinear Nonhomogeneous Obstacle Problems with Multivalued Convection Term

2022

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In this paper, a nonlinear elliptic obstacle problem is studied. The nonlinear nonhomogeneous partial differential operator generalizes the notions of p-Laplacian while on the right hand side we have a multivalued convection term (i.e., a multivalued reaction term may depend also on the gradient of the solution). The main result of the paper provides existence of the solutions as well as bondedness and closedness of the set of weak solutions of the problem, under quite general assumptions on the data. The main tool of the paper is the surjectivity theorem for multivalued functions given by the sum of a maximal monotone multivalued operator and a bounded multivalued pseudomonotone one.

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“…Neumann systems without gradient dependence on the nonlinearity can be found in Chabrowski [7] and de Godoi-Miyagaki-Rodrigues [11]. Finally, we mention some works pertaining equations exhibiting convection terms and subjected to Dirichlet or Neumann boundary conditions: we refer to Averna-Motreanu-Tornatore [1], de Araujo-Faria [10], Dupaigne-Ghergu-Rȃdulescu [13], El Manouni-Marino-Winkert [14], Faraci-Motreanu-Puglisi [16], Faraci-Puglisi [17], Figueiredo-Madeira [19], Gasiński-Papageorgiou [22], Gasiński-Winkert [23], Guarnotta-Marano-Motreanu [26], Liu-Motreanu-Zeng [30], Marano-Winkert [31], Motreanu-Tornatore [33], Motreanu-Winkert [34], Papageorgiou-Rȃdulescu-Repovš [35], and Vetro-Winkert [37].…”

Section: Introductionmentioning

confidence: 99%

The sub-supersolution method for variable exponent double phase systems with nonlinear boundary conditions

Guarnotta¹,

Livrea²,

Winkert³

2022

Preprint

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In this paper we study quasilinear elliptic systems driven by variable exponent double phase operators involving fully coupled right-hand sides and nonlinear boundary conditions. The aim of our work is to establish an enclosure and existence result for such systems by means of trapping regions formed by pairs of sup-and supersolutions. Under very general assumptions on the data we then apply our result to get infinitely many solutions. Moreover, we also discuss the case when we have homogeneous Dirichlet boundary conditions and present some existence results for this kind of problem.

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Positive maximal and minimal solutions for non-homogeneous elliptic equations depending on the gradient

Cited by 23 publications

References 56 publications

Regular solutions for nonlinear elliptic equations, with convective terms, in Orlicz spaces

Regular solutions for nonlinear elliptic equations, with convective terms, in Orlicz spaces

Nonlinear Nonhomogeneous Obstacle Problems with Multivalued Convection Term

The sub-supersolution method for variable exponent double phase systems with nonlinear boundary conditions

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