scholarly journals Harnack inequality and continuity of solutions for quasilinear elliptic equations in Sobolev spaces with variable exponent

2014 ◽  
Vol 2 ◽  
pp. 69-81 ◽  
Author(s):  
Azeddine Baalal ◽  
Abdelbaset Qabil
Keyword(s):  
Sobolev Spaces ◽  
Harnack Inequality ◽  
Elliptic Equations ◽  
Variable Exponent ◽  
Quasilinear Elliptic Equations ◽  
Continuity Of Solutions ◽  
Quasilinear Elliptic

scholarly journals Global W1,p(·) estimate for renormalized solutions of quasilinear equations with measure data on Reifenberg domains

2018 ◽  
Vol 7 (4) ◽  
pp. 517-533 ◽  
Author(s):  
The Anh Bui
Keyword(s):  
Elliptic Equations ◽  
Variable Exponent ◽  
Quasilinear Elliptic Equations ◽  
Measure Data ◽  
Renormalized Solutions ◽  
Lebesgue Spaces ◽  
Quasilinear Equations ◽  
Bmo Coefficients ◽  
Reifenberg Flat ◽  
Quasilinear Elliptic

AbstractIn this paper, we prove the gradient estimate for renormalized solutions to quasilinear elliptic equations with measure data on variable exponent Lebesgue spaces with BMO coefficients in a Reifenberg flat domain.

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.

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