# Weighted Choquard Equation Perturbed with Weighted Nonlocal Term

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### Article Info

DATE

2021-08-03

AUTHORS

Gurpreet Singh

ABSTRACT

We investigate the following problem -div(v(x)|∇u|m-2∇u)+V(x)|u|m-2u=|x|-θ∗|u|b|x|α|u|b-2|x|αu+λ|x|-γ∗|u|c|x|β|u|c-2|x|βuinRN,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}\begin{aligned} -\mathrm{div}(v(x)|\nabla u|^{m-2}\nabla u)+V(x)|u|^{m-2}u= \left( |x|^{-\theta }*\frac{|u|^{b}}{|x|^{\alpha }}\right) \frac{|u|^{b-2}}{|x|^{\alpha }}u+\lambda \left( |x|^{-\gamma }*\frac{|u|^{c}}{|x|^{\beta }}\right) \frac{|u|^{c-2}}{|x|^{\beta }}u \quad \text { in }{\mathbb {R}}^{N}, \end{aligned}\end{document}where b,c,α,β>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b, c, \alpha , \beta >0$$\end{document}, θ,γ∈(0,N)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta ,\gamma \in (0,N)$$\end{document}, N≥3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N\ge 3$$\end{document}, 2≤m<∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2\le m< \infty$$\end{document} and λ∈R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda \in {\mathbb {R}}$$\end{document}. Here, we are concerned with the existence of groundstate solutions and least energy sign-changing solutions and that will be done by using the minimization techniques on the associated Nehari manifold and the Nehari nodal set respectively. More... »

PAGES

1-21

### References to SciGraph publications

• 2020-06-05. Multiplicity of Solution for a Quasilinear Equation with Singular Nonlinearity in MEDITERRANEAN JOURNAL OF MATHEMATICS
• 2007-04-30. Problem with Critical Sobolev Exponent and with Weight in CHINESE ANNALS OF MATHEMATICS, SERIES B
• 2016-12-05. Choquard equations under confining external potentials in NONLINEAR DIFFERENTIAL EQUATIONS AND APPLICATIONS NODEA
• 2016-11-17. A guide to the Choquard equation in JOURNAL OF FIXED POINT THEORY AND APPLICATIONS
• 1996-05. On Gravity's role in Quantum State Reduction in GENERAL RELATIVITY AND GRAVITATION
• 2007. Measure Theory in NONE
• 2009-09-10. Semilinear elliptic equations with singular nonlinearities in CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS

### Identifiers

URI

http://scigraph.springernature.com/pub.10.1007/s12591-021-00579-3

DOI

http://dx.doi.org/10.1007/s12591-021-00579-3

DIMENSIONS

https://app.dimensions.ai/details/publication/pub.1140157462

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