Department of Mathematics
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Item 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.