Department of Mathematics
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Item Erratum to: Singular Adams inequality for biharmonic operator on Heisenberg Group and its applications(Springer, 2017-04) Dwivedi, GauravThe goal of this paper is to establish singular Adams type inequality for biharmonic operator on Heisenberg group. As an application, we establish the existence of a solution to where is a bounded domain, The special feature of this problem is that it contains an exponential nonlinearity and singular potential.Item Existence of multiple solutions for a kirchho¤ type equation Involving polyharmonic operator with exponential growth(2020-08) Dwivedi, GauravIn this article, we establish the existence of three weak solutions for a nonlinear Kirchho¤ type elliptic equation involving polyharmonic operator by using variational methods. We assume that the nonlinearity satis es subcritical exponential growth condition. We use a critical point theorem by B. Ricceri to prove our result.Item Generalised Picone's identity and some Qualitative properties of p-sub-Laplacian on Heisenberg group(DergiPark, 2021) Dwivedi, GauravIn this article, we derive a generalised nonlinear Picone's identity for p sub-Laplacian on the Heisenberg group. Our main result generalises the Picone's identity established by Niu et al.(Proceedings of the American Mathematical Society , Dec., 2001, Vol. 129, No. 12, pp. 3623-3630). As an application of Picone's identity, we prove a Hardy type inequality and Picone's inequality. We also establish some qualitative results involving the system of nonlinear equations involving p-sub-Laplacian.Item An existence result for -Laplace equation with gradient nonlinearity in R(EPI Sciences, 2022-05) Dwivedi, GauravWe prove the existence of a weak solution to the problem −Δpu+V(x)|u|p−2uu(x)=f(u,|∇u|p−2∇u), >0 ∀x∈RN, where Δpu=div(|∇u|p−2∇u) is the p-Laplace operator, 1Item Existence of solution to Kirchhoff type problem with gradient nonlinearity and a perturbation term(Springer, 2022-04) Dwivedi, GauravThis article deals with the existence of a weak solution to the Kirchhoff problem: where is a bounded and smooth domain in . We assume that f, h and A are continuous functions and the growth of the non linearity is dependent on u and . We do not assume any growth condition on the perturbation term h. In the case of we consider the exponential growth in the second variable of f. The proof of our main existence result uses an iterative technique based on the mountain pass theorem.Item Existence of solution to a nonlocal biharmonic problem with dependence on gradient and Laplacian(De Gruyter, 2022-01) Dwivedi, GauravIn this article, we prove the existence of a solution to a nonlocal biharmonic equation with nonlinearity depending on the gradient and the Laplacian. We employ an iterative technique based on the mountain pass theorem to prove our result.Item Ground state solution for a generalized Choquard Schrodinger equation with vanishing potential in homogeneous fractional Musielak Sobolev spaces(2023-01) Dwivedi, GauravThis paper aims to establish the existence of a weak solution for the following problem: (−Δ)sHu(x)+V(x)h(x,x,|u|)u(x)=(∫RNK(y)F(u(y))|x−y|λdy)K(x)f(u(x)) in RN, where N≥1, s∈(0,1),λ∈(0,N),H(x,y,t)=∫|t|0h(x,y,r)r dr, h:RN×RN×[0,∞)→[0,∞) is a generalized N-function and (−Δ)sH is a generalized fractional Laplace operator. The functions V,K:RN→(0,∞), non-linear function f:R→R are continuous and F(t)=∫t0f(r)dr. First, we introduce the homogeneous fractional Musielak-Sobolev space and investigate their properties. After that, we pose the given problem in that space. To establish our existence results, we prove and use the suitable version of Hardy-Littlewood-Sobolev inequality for Lebesque Musielak spaces together with variational technique based on the mountain pass theorem. We also prove the existence of a ground state solution by the method of Nehari manifold.Item Biharmonic elliptic problems with second Hessian and gradient nonlinearities(Wiley, 2022-11) Dwivedi, GauravWe establish the existence of a solution to the following problem: where , is a smooth and bounded domain and , where is the ith eigenvalue of symmetric matrix . We assume that and are parameters. Moreover, we assume that if and if . We use variational arguments and an iterative technique to prove our results.Item Ground state solution to n-kirchhoff equation with critical exponential growth and without ambrosetti–rabinowitz condition(Springer, 2023-05) Dwivedi, GauravThis article is focused on the existence of a ground state solution to the Kirchhoff problem: where is a bounded domain with smooth boundary and . We assume that f satisfies critical exponential growth at infinity but does not satisfy the well-known Ambrosetti–Rabinowitz condition. We prove the existence of a ground state weak solution via mountain pass theorem and Nehari manifold technique.Item Kirchhoff type elliptic equations with double criticality in Musielak–Sobolev spaces(Wiley, 2023-01) Dwivedi, GauravThis paper aims to establish the existence of a weak solution for the nonlocal problem: where is a bounded and smooth domain containing two open and connected subsets and such that and is the -Laplace operator. We assume that reduces to in and to in , the nonlinear function acts as on and as on for sufficiently large . To establish the existence results in a Musielak–Sobolev space, we use a variational technique based on the mountain pass theorem.
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