scholarly journals Double phase problems with variable growth and convection for the Baouendi–Grushin operator

2020 ◽  
Vol 71 (6) ◽  
Author(s):  
Anouar Bahrouni ◽  
Vicenţiu D. Rădulescu ◽  
Patrick Winkert
Keyword(s):  
Elliptic Equations ◽  
Quasilinear Elliptic Equations ◽  
Stationary Waves ◽  
Convection Term ◽  
Pseudomonotone Operators ◽  
Qualitative Properties ◽  
Transonic Flows ◽  
Grushin Operator ◽  
Double Phase ◽  
Quasilinear Elliptic

AbstractIn this paper we study a class of quasilinear elliptic equations with double phase energy and reaction term depending on the gradient. The main feature is that the associated functional is driven by the Baouendi–Grushin operator with variable coefficient. This partial differential equation is of mixed type and possesses both elliptic and hyperbolic regions. We first establish some new qualitative properties of a differential operator introduced recently by Bahrouni et al. (Nonlinearity 32(7):2481–2495, 2019). Next, under quite general assumptions on the convection term, we prove the existence of stationary waves by applying the theory of pseudomonotone operators. The analysis carried out in this paper is motivated by patterns arising in the theory of transonic flows.

Singular nonhomogeneous quasilinear elliptic equations with a convection term

2017 ◽  
Vol 290 (14-15) ◽  
pp. 2280-2295
Author(s):  
J. V. Gonçalves ◽  
M. R. Marcial ◽  
O. H. Miyagaki
Keyword(s):  
Elliptic Equations ◽  
Quasilinear Elliptic Equations ◽  
Convection Term ◽  
Quasilinear Elliptic

scholarly journals Multiple solutions of double phase variational problems with variable exponent

2020 ◽  
Vol 13 (4) ◽  
pp. 385-401 ◽  
Author(s):  
Xiayang Shi ◽  
Vicenţiu D. Rădulescu ◽  
Dušan D. Repovš ◽  
Qihu Zhang
Keyword(s):  
Elliptic Equations ◽  
Multiple Solutions ◽  
Variable Exponent ◽  
Quasilinear Equation ◽  
Variational Problems ◽  
Quasilinear Elliptic Equations ◽  
Weighting Method ◽  
Double Phase ◽  
Unbounded Potential ◽  
Quasilinear Elliptic

AbstractThis paper deals with the existence of multiple solutions for the quasilinear equation{-\operatorname{div}\mathbf{A}(x,\nabla u)+|u|^{\alpha(x)-2}u=f(x,u)\quad\text% {in ${\mathbb{R}^{N}}$,}}which involves a general variable exponent elliptic operator {\mathbf{A}} in divergence form. The problem corresponds to double phase anisotropic phenomena, in the sense that the differential operator has various types of behavior like {|\xi|^{q(x)-2}\xi} for small {|\xi|} and like {|\xi|^{p(x)-2}\xi} for large {|\xi|}, where {1<\alpha(\,\cdot\,)\leq p(\,\cdot\,)<q(\,\cdot\,)<N}. Our aim is to approach variationally the problem by using the tools of critical points theory in generalized Orlicz–Sobolev spaces with variable exponent. Our results extend the previous works [A. Azzollini, P. d’Avenia and A. Pomponio, Quasilinear elliptic equations in \mathbb{R}^{N} via variational methods and Orlicz–Sobolev embeddings, Calc. Var. Partial Differential Equations 49 2014, 1–2, 197–213] and [N. Chorfi and V. D. Rădulescu, Standing wave solutions of a quasilinear degenerate Schrödinger equation with unbounded potential, Electron. J. Qual. Theory Differ. Equ. 2016 2016, Paper No. 37] from cases where the exponents p and q are constant, to the case where {p(\,\cdot\,)} and {q(\,\cdot\,)} are functions. We also substantially weaken some of the hypotheses in these papers and we overcome the lack of compactness by using the weighting method.

On a Class of Quasilinear Elliptic Equations with Degenerate Coerciveness and Measure Data

2018 ◽  
Vol 18 (2) ◽  
pp. 361-392 ◽  
Author(s):  
Flavia Smarrazzo
Keyword(s):  
Elliptic Equations ◽  
Quasilinear Elliptic Equations ◽  
Measure Data ◽  
Qualitative Properties ◽  
Uniqueness Results ◽  
Diffuse Part ◽  
Degenerate Elliptic ◽  
Harmonic Capacity ◽  
Quasilinear Elliptic ◽  
Standard Approximation

AbstractWe study the existence of measure-valued solutions for a class of degenerate elliptic equations with measure data. The notion of solution is natural, since it is obtained by a regularization procedure which also relies on a standard approximation of the datum μ. We provide partial uniqueness results and qualitative properties of the constructed solutions concerning, in particular, the structure of their diffuse part with respect to the harmonic-capacity.

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