Gurpreet Singh

Bio: Gurpreet Singh is an academic researcher from University College Dublin. The author has an hindex of 1, co-authored 1 publications receiving 16 citations.

Nonlocal perturbations of the fractional Choquard equation

Abstract: Abstract We study the equation ( - Δ ) s ⁢ u + V ⁢ ( x ) ⁢ u = ( I α * | u | p ) ⁢ | u | p - 2 ⁢ u + λ ⁢ ( I β * | u | q ) ⁢ | u | q - 2 ⁢ u in ⁢ ℝ N , (-\\Delta)^{s}u+V(x)u=(I_{\\alpha}*\\lvert u\\rvert^{p})\\lvert u\\rvert^{p-2}u+% \\lambda(I_{\\beta}*\\lvert u\\rvert^{q})\\lvert u\\rvert^{q-2}u\\quad\\text{in }{% \\mathbb{R}}^{N}, where I γ ⁢ ( x ) = | x | - γ {I_{\\gamma}(x)=\\lvert x\\rvert^{-\\gamma}} for any γ ∈ ( 0 , N ) {\\gamma\\in(0,N)} , p , q > 0 {p,q>0} , α , β ∈ ( 0 , N ) {\\alpha,\\beta\\in(0,N)} , N ≥ 3 {N\\geq 3} , and λ ∈ ℝ {\\lambda\\in{\\mathbb{R}}} . First, the existence of groundstate solutions by using a minimization method on the associated Nehari manifold is obtained. Next, the existence of least energy sign-changing solutions is investigated by considering the Nehari nodal set.

On a logarithmic Hartree equation

Abstract: Abstract We study the existence of radially symmetric solutions for a nonlinear planar Schrödinger-Poisson system in presence of a superlinear reaction term which doesn’t satisfy the Ambrosetti-Rabinowitz condition. The system is re-written as a nonlinear Hartree equation with a logarithmic convolution term, and the existence of a positive and a negative solution is established via critical point theory.

Ground states and non-existence results for Choquard type equations with lower critical exponent and indefinite potentials

Abstract: In this paper, we study following Choquard equation with lower critical exponent: − Δ u + V ( x ) u = ( I α ∗ | u | α N + 1 ) | u | α N − 1 u + f ( x , u ) , x ∈ R N , u ∈ H 1 ( R N ) , where N ≥ 1 , I α is the Riesz potential, and V : R N → R allows to be sign-changing. Under some mild assumptions imposed on the nonlinearity f , we prove that above equation has a ground state solution for the periodic case and asymptotically periodic case, respectively. The characterization of the ground states is also investigated by a direct approach that are constrained to the Nehari manifold. Moreover, we show a non-existence result for the equation via a generalized Pohožaev identity established for the non-autonomous nonlinearity f . The results extend and improve some recent ones in the literature.

Bifurcation results for the critical Choquard problem involving fractional p-Laplacian operator

Abstract: By using an abstract critical point theorem based on a pseudo-index related to the cohomological index, we prove the bifurcation results for the critical Choquard problems involving fractional p-Laplacian operator: $$(- \Delta)_{p}^{s} u = \lambda \vert u \vert ^{p-2} u + \biggl( \int_{\Omega}\frac{ \vert u \vert ^{p_{\mu,s}^{*}}}{ \vert x-y \vert ^{\mu}}\,dy \biggr) \vert u \vert ^{p_{\mu,s}^{*}-2}u \quad \text{in } \Omega,\qquad u = 0\quad \text{in } {\mathbb {R}}^{N} \setminus \Omega, $$ where Ω is a bounded domain in ${\mathbb {R}}^{N}$ with Lipschitz boundary, λ is a real parameter, $p\in(1,\infty)$ , $s\in (0,1)$ , $N>sp$ , and $p_{\mu,s}^{*}=\frac{(N-\frac{\mu}{2})p}{N-sp}$ is the critical exponent in the sense of Hardy–Littlewood–Sobolev inequality. These extend results in the literature for the fractional Choquard problems, and they are still new for a p-Laplacian case.

A class of elliptic equations involving nonlocal integrodifferential operators with sign-changing weight functions

Abstract: In this paper, we investigate the existence of nontrivial weak solutions to a class of elliptic equations involving a general nonlocal integrodifferential operator LAK, two real parameters, and two weight functions, which can be sign-changing. Considering different situations concerning the growth of the nonlinearities involved in the problem (P), we prove the existence of two nontrivial distinct solutions and the existence of a continuous family of eigenvalues. The proofs of the main results are based on ground state solutions using the Nehari method, Ekeland’s variational principle, and the direct method of the calculus of variations. The difficulties arise from the fact that the operator LAK is nonhomogeneous and the nonlinear term is undefined.