Multiplicity and asymptotic behavior of solutions for Kirchhoff type equations involving the Hardy\u2013Sobolev exponent and singular nonlinearity

Liejun Shen

doi:10.1186/s13660-018-1806-8

Abstract

In this paper, we study a class of critical elliptic problems of Kirchhoff type: \t\t\t[a+b(∫R3|∇u|2−μu2|x|2dx)2−α2](−Δu−μu|x|2)=|u|2∗(α)−2u|x|α+λf(x)|u|q−2u|x|β,\\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}$$ \\biggl[a+b \\biggl( \\int_{\\mathbb{R}^{3}}\\vert \\nabla u\\vert ^{2}-\\mu \\frac{u^{2}}{\\vert x\\vert ^{2}}\\,dx \\biggr)^{\\frac{2-\\alpha }{2}} \\biggr]\\biggl(-\\Delta u- \\mu \\frac{u}{\\vert x\\vert ^{2}}\\biggr) = \\frac{\\vert u\\vert ^{2^{*}(\\alpha )-2}u }{\\vert x\\vert ^{\\alpha }}+\\lambda \\frac{f(x)\\vert u\\vert ^{q-2}u }{\\vert x\\vert ^{\\beta }}, $$\\end{document} where a,b>0, mu in [0,1/4), alpha , beta in [0,2), and qin (1,2) are constants and 2^{*}(alpha )=6-2alpha is the Hardy–Sobolev exponent in mathbb{R}^{3}. For a suitable function f(x), we establish the existence of multiple solutions by using the Nehari manifold and fibering maps. Moreover, we regard b>0 as a parameter to obtain the convergence property of solutions for the given problem as bsearrow 0^{+} by the mountain pass theorem and Ekeland’s variational principle.

Highlights

They obtained the existence and multiplicity of positive solutions to the above problem
Introduction and main resultsIn the present paper, we consider the following Schrödinger equation: |∇ u|2 μ u2 |x|2 dx|u|2∗(α)–2u f (x)|u|q–2u = |x|α + λ |x|β, u – u – μ |x|2 (1.1)where a, b > 0, μ ∈ [0, 1/4), α, β ∈ [0, 2), and q ∈ (1, 2) are constants and 2∗(α) = 6 – 2α is the critical Hardy–Sobolev exponent.We call (1.1) a Schrödinger equation of Kirchhoff type because of the appearance of the term b( R3 |∇u|2 – μu2|x|–2 dx)(2–α)/2 which makes the study of (1.1) interesting
In this paper, we study a class of critical elliptic problems of Kirchhoff type: a+b

Summary

Introduction

They obtained the existence and multiplicity of positive solutions to the above problem. ⎧ ⎨–M( |∇u|2 dx) u = g(x)|u|p–2u + λh(x)|u|q–2u in , ⎩u = 0 on ∂ , where M is the so-called Kirchhoff function depending on 1 < q < 2 < p < 2∗, is a bounded domain with a smooth boundary in RN and the weight functions h, g ∈ C( ) satisfy some specified conditions f ± = max{±f , 0} = 0 and g± = max{±g, 0} = 0, they proved the existence of multiple solutions of it. Let λ ∈ (0, λM) and a > 0 be fixed constants, there exist subsequences still denoted by themselves {u1b} and {u2b} such that uib → ui in D1,2(R3) as b 0+ for i ∈ {1, 2}, where u1 and u2 are two nontrivial solutions of u

Results

Conclusion