Abstract
The paper deals with the following Kirchhoff–Poisson systems: \t\t\t0.1{−(1+b∫R3|∇u|2dx)Δu+u+k(x)ϕu+λ|u|p−2u=h(x)|u|q−2u,x∈R3,−Δϕ=k(x)u2,x∈R3,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\textstyle\\begin{cases} - ( {1+b\\int _{{\\mathbb{R}}^{3}} { \\vert \\nabla u \\vert ^{2}\\,dx} } ) \\Delta u+u+k(x)\\phi u+\\lambda \\vert u \\vert ^{p-2}u=h(x) \\vert u \\vert ^{q-2}u, & x\\in {\\mathbb{R}}^{3}, \\\\ -\\Delta \\phi =k(x)u^{2}, & x\\in {\\mathbb{R}}^{3}, \\end{cases} $$\\end{document} where the functions k and h are nonnegative, 0le lambda , b; 2le ple 4< q<6. Via a constraint variational method combined with a quantitative lemma, some existence results on one least energy sign-changing solution with two nodal domains to the above systems are obtained. Moreover, the convergence property of u_{b} as b searrow 0 is established.
Highlights
Consider the following Kirchhoff–Poisson systems:⎧ ⎨–(1 + b R3 |∇u|2 dx) u + u + k(x)φu + λ|u|p–2u = h(x)|u|q–2u, ⎩– φ = k(x)u2, x ∈ R3, x ∈ R3, (1.1)where k and h are nonnegative functions, 0 ≤ λ, b; 2 ≤ p ≤ 4 < q < 6
⎩– φ = k(x)u2, (1.2) x ∈ R3, which stem from quantum mechanics and have important applications in the semiconductor
Theorem 3.1 Assume that conditions (l), (k), (h) hold, problem (1.1) possesses one least energy sign-changing solution ub, which has exactly two nodal domains, where ub is given in Lemma 2.6
Summary
The authors obtained the existence results on the ground state sign-changing solution for 0 < λ < λ1. Shuai in [28] studied the existence of the least energy sign-changing solution of problem (1.5) and its convergence property on {un} as b 0. Under conditions different from [28], with the help of some analytical techniques and a non-Nehari manifold method, Tang and Cheng in [29] further studied problem (1.5) and obtained some existence results on a ground state sign-changing solution ub as well as its convergence property. Different from the works mentioned above, in the present paper, we shall combine a constraint variational method with quantitative deformation properties to establish the existence results as regards one least energy sigh-changing solution with two nodal domains to problem (1.1). We study the convergence property on ub as b 0
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
