### Browsing by Author "Garain, Prashanta"

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Item Existence and nonexistence results for anisotropic p-Laplace equation with singular nonlinearities(TAYLOR & FRANCIS, 2020-08-05) Garain, Prashanta; Department of Mathematics and Systems AnalysisLet pi≥ 2 and consider the following anisotropic p-Laplace equation −∑Ni=1∂∂xi(∣∣∂u∂xi∣∣pi−2∂u∂xi)=g(x)f(u),u>0 in Ω. Under suitable hypothesis on the weight function g we present an existence result for f(u)=e1u in a bounded smooth domain Ω and nonexistence results for f(u)=−e1u or −(u−δ+u−γ), δ,γ>0 with Ω=RN respectively.Item High perturbations of quasilinear problems with double criticality(SPRINGER HEIDELBERG, 2021-12) Alves, Claudianor O.; Garain, Prashanta; Rădulescu, Vicenţiu D.; Department of Mathematics and Systems Analysis; Universidade Federal de Campina Grande; AGH University of Science and TechnologyThis paper is concerned with the qualitative analysis of solutions to the following class of quasilinear problems {-ΔΦu=f(x,u)inΩ,u=0on∂Ω,where ΔΦu=div(φ(x,|∇u|)∇u) and Φ(x,t)=∫0|t|φ(x,s)sds is a generalized N-function. We assume that Ω ⊂ RN is a smooth bounded domain that contains two open regions Ω N, Ω p with Ω ¯ N∩ Ω ¯ p= ∅. The features of this paper are that - Δ Φu behaves like - Δ Nu on Ω N and - Δ pu on Ω p, and that the growth of f: Ω × R→ R is like that of eα|t|NN-1 on Ω N and as |t|p∗-2t on Ω p when |t| is large enough. The main result establishes the existence of solutions in a suitable Musielak–Sobolev space in the case of high perturbations with respect to the values of a positive parameter.Item Lower semicontinuity and pointwise behavior of supersolutions for some doubly nonlinear nonlocal parabolic p-Laplace equations(WORLD SCIENTIFIC, 2023-10-01) Banerjee, Agnid; Garain, Prashanta; Kinnunen, Juha; Department of Mathematics and Systems Analysis; Analysis; Tata Institute of Fundamental Research; Ben-Gurion University of the NegevWe discuss pointwise behavior of weak supersolutions for a class of doubly nonlinear parabolic fractional p-Laplace equations which includes the fractional parabolic p-Laplace equation and the fractional porous medium equation. More precisely, we show that weak supersolutions have lower semicontinuous representative. We also prove that the semicontinuous representative at an instant of time is determined by the values at previous times. This gives a pointwise interpretation for a weak supersolution at every point. The corresponding results hold true also for weak subsolutions. Our results extend some recent results in the local parabolic case, and in the nonlocal elliptic case, to the nonlocal parabolic case. We prove the required energy estimates and measure theoretic De Giorgi type lemmas in the fractional setting.Item Properties of solutions to some weighted p-Laplacian equation(2020) Garain, Prashanta; Pucci, Patrizia; Department of Mathematics and Systems AnalysisIn this paper, we prove some qualitative properties for the positive solutions to some degenerate elliptic equation given by-div (w|∇u|p-2∇u) = f(x, u), u ∈ Ap, on smooth domain and for varying nonlinearity f.Item Some local properties of subsolution and supersolutions for a doubly nonlinear nonlocal p-Laplace equation(Springer Verlag, 2022) Banerjee, Agnid; Garain, Prashanta; Kinnunen, Juha; Department of Mathematics and Systems Analysis; Analysis; Tata Institute of Fundamental ResearchWe establish a local boundedness estimate for weak subsolutions to a doubly nonlinear parabolic fractional p-Laplace equation. Our argument relies on energy estimates and a parabolic nonlocal version of De Giorgi’s method. Furthermore, by means of a new algebraic inequality, we show that positive weak supersolutions satisfy a reverse Hölder inequality. Finally, we also prove a logarithmic decay estimate for positive supersolutions.Item Weak Harnack inequality for a mixed local and nonlocal parabolic equation(Academic Press, 2023-07-05) Garain, Prashanta; Kinnunen, Juha; Department of Mathematics and Systems Analysis; AnalysisThis article proves a weak Harnack inequality with a tail term for sign changing supersolutions of a mixed local and nonlocal parabolic equation. Our argument is purely analytic. It is based on energy estimates and the Moser iteration technique. Instead of the parabolic John-Nirenberg lemma, we adopt a lemma of Bombieri-Giusti to the mixed local and nonlocal parabolic case. To this end, we prove an appropriate reverse Hölder inequality and a logarithmic estimate for weak supersolutions.