Department of Mathematics
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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.