Department of Mathematics
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Item Existence and multiplicity of solutions to N-Kirchhoff equations with critical exponential growth and a perturbation term(Taylor & Francis, 2022-03) Dwivedi, GauravThe aim of this article is twofold: firstly, we deal with the existence and multiplicity of weak solutions to the Kirchhoff problem: ⎧ ⎪ ⎨ ⎪ ⎩ −𝑎(∫Ω|∇𝑢|𝑁d𝑥)Δ𝑁𝑢= 𝑓(𝑥,𝑢) |𝑥|𝑏 +𝜆ℎ(𝑥)in Ω,𝑢=0on ∂Ω, where Ω is a smooth bounded domain in ℝ𝑁(𝑁≥ 2) and 0≤𝑏<𝑁. Secondly, we deal with the existence and multiplicity of weak solutions to the Kirchhoff problem: −𝑎(∫ℝ𝑁|∇𝑢|𝑁+𝑉(𝑥)|𝑢|𝑁d𝑥)(Δ𝑁𝑢+𝑉(𝑥)|𝑢|𝑁−2𝑢)= 𝑔(𝑥,𝑢) |𝑥|𝑏 +𝜆ℎ(𝑥)in ℝ𝑁, where 𝑁≥ 2 and 0≤𝑏<𝑁. We assume that f and g have critical exponential growth at infinity. To establish our existence results, we use the mountain pass theorem, Ekeland variational principle and Moser–Trudinger inequality.Item Stability of Positive Solution to Fractional Logistic Equations(Division of Functional Equations, 2019) Dwivedi, GauravIn this paper, we show the existence of a classical solution to a class of fractional logistic equations in an open bounded subset with smooth boundary. We use the method of sub- and super-solutions with variational arguments to establish the existence of a unique positive solution. We also establish the stability and nondegeneracy of the positive solution.Item Picone’s identity for biharmonic operators on Heisenberg group and its applications(Springer, 2016) Dwivedi, GauravIn this paper, we establish a nonlinear analogue of Picone’s identity for biharmonic operators on Heisenberg group. As an applications of Picone’s identity, we obtain Hardy-Rellich type inequality, Morse index, Caccioppoli inequality, Picone inequality for biharmonic operators on Heisenberg group