Browsing by Author "Dwivedi, Gaurav"

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Biharmonic elliptic problems with second Hessian and gradient nonlinearities
(Wiley, 2022-11) Dwivedi, Gaurav
We 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.
Erratum to: Singular Adams inequality for biharmonic operator on Heisenberg Group and its applications
(Springer, 2017) Dwivedi, Gaurav
We have established Adams-type inequality for biharmonic operator on Heisenberg group and proved the existence of solution to a biharmonic equation involving a singular potential and a nonlinearity satisfying critical and subcritical exponential growth condition. We observed that there is a technical mistake in the homogeneous dimension of the Heisenberg group that is under consideration. For our results to be meaningful, we need to work with bounded domains in H1 instead of bounded domains in H4. The reason of this change is as follows: Let Ω ⊆ Hn be a bounded domain and Q = 2n + 2 be homogeneous dimension of Hn. When Q > 4 (n > 1), we know that D2,2 0 (Ω) → Lq(Ω), 1 ≤ q ≤ 2Q Q−4 . In the critical case, Q = 4(n = 1), D2,2 0 (Ω) → L∞(Ω). Then it is natural to ask, what is the best possible space for this embedding? To answer this question, we need an Adams-type inequality with Q = 4. Thus, we need to work with H1 instead of H4 in [1]. For the sake of clarity, we restate the main results of [1]. However, all the proofs remain unchanged.
Erratum to: Singular Adams inequality for biharmonic operator on Heisenberg Group and its applications
(Springer, 2017-04) Dwivedi, Gaurav
The 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.
Existence and multiplicity of solutions to N-Kirchhoff equations with critical exponential growth and a perturbation term
(Taylor & Francis, 2022-03) Dwivedi, Gaurav
The 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)⁢Δ𝑁⁢𝑢= 𝑓⁡(𝑥,𝑢) |𝑥|𝑏 +𝜆⁢ℎ⁡(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)⁢(Δ𝑁⁢𝑢+𝑉⁡(𝑥)⁢|𝑢|𝑁−2⁢𝑢)= 𝑔⁡(𝑥,𝑢) |𝑥|𝑏 +𝜆⁢ℎ⁡(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.
Existence of multiple solutions for a kirchho¤ type equation Involving polyharmonic operator with exponential growth
(2020-08) Dwivedi, Gaurav
In 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.
Existence of solution for biharmonic systems with indefinite weights
(Ele-math, 2014) Dwivedi, Gaurav
In this article we deal with the existence questions to the nonlinear biharmonic systems. Using theory of monotone operators, we show the existence of a unique weak solution to the weighted biharmonic systems. We also show the existence of a positive solution to weighted biharmonic systems in the unit ball in Rn , using Leray Schauder fixed point theorem. In this study we allow sign-changing weights
Existence of solution to a nonlocal biharmonic problem with dependence on gradient and Laplacian
(De Gruyter, 2022-01) Dwivedi, Gaurav
In 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.
Existence of solution to Kirchhoff type problem with gradient nonlinearity and a perturbation term
(Springer, 2022-04) Dwivedi, Gaurav
This 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.
Existence of weak solutions for Kirchhoff type double-phase problem in R
(SAO Astrophysic, 2024) Dwivedi, Gaurav
This article aims to study the existence of weak solutions for the Kirchhoff type double-phase problem imagewhere and is a real parameter. The main distinctive feature of this problem lies in the second nonlinearity on the right-hand side, which can be in the supercritical. Additionally, we encounter the Kirchhoff term, which might become zero at the origin. By imposing certain assumptions, we establish the existence and multiplicity results.
An existence result for -Laplace equation with gradient nonlinearity in R
(EPI Sciences, 2022-05) Dwivedi, Gaurav
We 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, 1
Existence results for singular double phase problem with variable exponents
(Springer, 2023) Dwivedi, Gaurav
The goal of this paper is to prove the existence of two weak solutions for the following problem: − div(|∇u|p(x)−2∇u+μ(x)|∇u|q(x)−2∇u) =λh(x)u −η(x)+ξ(x)|u|s(x)−2u in Ω, u = 0 on ∂Ω, where Ω ⊂ RN, N ≥ 2 is a bounded domain with smooth boundary ∂Ω and λ > 0 is a real parameter. The functions h(x), ξ(x) ∈ C(Ω) are positive with compact support in Ω.We assume some suitable conditions on functions p, q, η and s. We use the Nehari manifold method based on fibering maps to establish our results. Mathematics Subject Classification. 35J30, 35J75, 35D30. Keywords. Nehari manifold,Musielak–Sobolev spaces, fibering map, double phase operator with variable exponents
Generalised Picone's identity and some Qualitative properties of p-sub-Laplacian on Heisenberg group
(DergiPark, 2021) Dwivedi, Gaurav
In 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.
Ground state solution for a generalized Choquard Schrodinger equation with vanishing potential in homogeneous fractional Musielak Sobolev spaces
(2023-01) Dwivedi, Gaurav
This 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.
Ground state solution to n-kirchhoff equation with critical exponential growth and without ambrosetti–rabinowitz condition
(Springer, 2023-05) Dwivedi, Gaurav
This 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.
Kirchhoff type elliptic equations with double criticality in Musielak–Sobolev spaces
(Wiley, 2023-01) Dwivedi, Gaurav
This 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.
A Note on the Caccioppoli Inequality for Biharmonic Operators
(Springer, 2015-07) Dwivedi, Gaurav
In this note, we establish a Caccioppoli-type inequality for biharmonic operators. We employ Picone’s identity for biharmonic operators to establish Caccioppoli-type inequality.
On the bifurcation results for fractional Laplace equations
(Wiley, 2017-02) Dwivedi, Gaurav
In this paper, we consider the bifurcation problem for the fractional Laplace equation urn:x-wiley:0025584X:media:mana201600250:mana201600250-math-0001 where urn:x-wiley:0025584X:media:mana201600250:mana201600250-math-0002 is an open bounded subset with smooth boundary, urn:x-wiley:0025584X:media:mana201600250:mana201600250-math-0003 stands for the fractional Laplacian. We show that a continuum of solutions bifurcates out from the principal eigenvalue λ1 of the problem urn:x-wiley:0025584X:media:mana201600250:mana201600250-math-0004 and, conversely.
Picone’s identity for biharmonic operators on Heisenberg group and its applications
(Springer, 2016) Dwivedi, Gaurav
In 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
Picone's Identity for p-biharmonic operator and Its Applications
(ARXIV, 2015-03) Dwivedi, Gaurav
In this article we prove the nonlinear analogue of Picone's identity for p−biharmonic operator. As an application of our result we show that the Morse index of the zero solution to a p−biharmonic boundary value problem is 0. We also prove a Hardy type inequality and Sturmian comparison principle. We also show the strict monotonicity of the principle eigenvalue and linear relationship between the solutions of a system of singular p-biharmonic system.
Some qualitative questions on the equation −div(a(x,u,∇u))=f(x, u)
(Elsevier, 2017-02) Dwivedi, Gaurav
In this article, we establish several applications of Picone's identity for the operator of the form such as Hardy type inequality, Sturmian comparison theorem, monotonicity property of the first eigenvalue, nonexistence of positive supersolutions and Caccioppoli inequality under certain conditions on a.