Double phase anisotropic variational problems and combined effects of reaction and absorption terms

Zhang, Qihu; Rădulescu, Vicenţiu D.

doi:10.1016/j.matpur.2018.06.015

Cited by 131 publications

(59 citation statements)

References 45 publications

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“…and we denote by v 1 << v 2 the fact that From Lemma A.2, given in Appendix A of [61], it is clear that this typical potential A satisfies (A 1 ) and (A 2 ), 1 < p − ≤ p + < N and 1 < q − ≤ q + < N .…”

Section: Introductionmentioning

confidence: 99%

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Multiple solutions of double phase variational problems with variable exponent

Shi

Rădulescu

Repovš

et al. 2018

Advances in Calculus of Variations

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This 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 exponentspandqare 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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Section: Introductionmentioning

confidence: 99%

“…Then (X(Ω), u Ω ) is a Banach space.Proposition 3.7. (see[61, Proposition 3.11]) Assume (A 1 )-(iv). Then (X(Ω), u Ω ) is reflexive.…”

mentioning

confidence: 99%

Multiple solutions of double phase variational problems with variable exponent

Shi

Rădulescu

Repovš

et al. 2018

Advances in Calculus of Variations

Self Cite

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“…We change the exponents of both two absorption sources such that the two terms have a large enough gaps in the sense of growth order, in order to reveal and compare the importance of the two factors acting on the behavior of the quenching phenomena, which are the number of the absorption source terms and the exponents of these terms. The results obtained in the preset paper suggests the dominant influence of the exponents of the absorption terms comparing the number of them, which is not only different from the classical heat equation with nonlinear power-type external force [12,13], but also different from the nonlinearities and their corresponding behaviours and affects in other models [14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30].…”

Section: Introductionmentioning

confidence: 99%

Quenching Time Estimates for Semilinear Parabolic Equations Controlled by Two Absorption Sources in Control System

sci¹

2020

JPDE

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This paper deals with the quenching solution of the initial boundary value problem for aclass of semilinear reaction-diffusion equation controlled by two absorption sources in control system and estimate upper bound and lower bound of the quenching time. We point that the number of absorption sources influences the time of quenching phenomenon.The solution can solve some boundary value problem in control system.

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“…In this paper, the double degenerate evolutionary p(x)-Laplacian equation u t = div b(x, t) ∇A(u) p(x)-2 ∇A(u) + f (x, t, u, ∇u), (x, t) ∈ Q T = Ω × (0, T), (1.1) is considered, in which Ω ⊂ R N is a bounded domain with smooth boundary ∂Ω, p(x) > 1 is a C 1 [1,2]), and has been widely studied [2][3][4][5][6][7][8][9][10][11][12][13][14][15] in recent decade. If b(x, t) = 1, p(x) = p > 1 is a constant, equation (1.1) is a generalization of the following polytropic infiltration equation:…”

Section: Introductionmentioning

confidence: 99%

On a weak solution matching up with the double degenerate parabolic equation

Weng

2019

Bound Value Probl

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The well-posedness of weak solutions to a double degenerate evolutionary p(x)-Laplacian equation u t = div(b(x, t) ∇A(u) p(x)-2 ∇A(u)), is studied. It is assumed that b(x, t)| (x,t)∈Ω×[0,T] > 0 but b(x, t)| (x,t)∈∂Ω×[0,T] = 0, A (s) = a(s) ≥ 0, and A(s) is a strictly monotone increasing function with A(0) = 0. A weak solution matching up with the double degenerate parabolic equation is introduced. The existence of weak solution is proved by a parabolically regularized method. The stability theorem of weak solutions is established independent of the boundary value condition. In particular, the initial value condition is satisfied in a wider generality.

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Double phase anisotropic variational problems and combined effects of reaction and absorption terms

Cited by 131 publications

References 45 publications

Multiple solutions of double phase variational problems with variable exponent

Multiple solutions of double phase variational problems with variable exponent

Quenching Time Estimates for Semilinear Parabolic Equations Controlled by Two Absorption Sources in Control System

On a weak solution matching up with the double degenerate parabolic equation

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