Abstract
In this paper, we are concerned with the existence of least energy sign-changing solutions for the following fractional Kirchhoff problem with logarithmic and critical nonlinearity: a+b[u]s,pp(-Δ)psu=λ|u|q-2uln|u|2+|u|ps∗-2uinΩ,u=0inRN\\Ω,\\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}$$\\begin{aligned} \\left\\{ \\begin{array}{ll} \\left( a+b[u]_{s,p}^p\\right) (-\\Delta )^s_pu = \\lambda |u|^{q-2}u\\ln |u|^2 + |u|^{ p_s^{*}-2 }u &{}\\quad \\text {in } \\Omega , \\\\ u=0 &{}\\quad \\text {in } {\\mathbb {R}}^N{\\setminus } \\Omega , \\end{array}\\right. \\end{aligned}$$\\end{document}where N >sp with s in (0, 1), p>1, and [u]s,pp=∬R2N|u(x)-u(y)|p|x-y|N+psdxdy,\\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}$$\\begin{aligned}{}[u]_{s,p}^p =\\iint _{{\\mathbb {R}}^{2N}}\\frac{|u(x)-u(y)|^p}{|x-y|^{N+ps}}dxdy, \\end{aligned}$$\\end{document}p_s^*=Np/(N-ps) is the fractional critical Sobolev exponent, Omega subset {mathbb {R}}^N(Nge 3) is a bounded domain with Lipschitz boundary and lambda is a positive parameter. By using constraint variational methods, topological degree theory and quantitative deformation arguments, we prove that the above problem has one least energy sign-changing solution u_b. Moreover, for any lambda > 0, we show that the energy of u_b is strictly larger than two times the ground state energy. Finally, we consider b as a parameter and study the convergence property of the least energy sign-changing solution as b rightarrow 0.
Highlights
We are interested in the existence, energy estimates and the convergence property of the least energy sign-changing solution for the following fractional Kirchhoff problems with logarithmic and critical nonlinearity: a + b[u]sp,p (− )spu = λ|u|q−2u ln |u|2 + |u|ps∗−2u in, u=0 in RN \, (1.1)
In [43], the authors obtained the existence of least energy sign-changing solutions of Kirchhoff-type equation with critical growth by using the constraint variational method and the quantitative deformation lemma
To the best of our knowledge, there are no results concerning the existence of signchanging solutions for fractional Kirchhoff problems with logarithmic and critical nonlinearity
Summary
We are interested in the existence, energy estimates and the convergence property of the least energy sign-changing solution for the following fractional Kirchhoff problems with logarithmic and critical nonlinearity:. In [42], Truong studied the following problem fractional p-Laplacian equations with logarithmic nonlinearity (− )spu + V (x)|u|p−2u = λa(x)|u|p−2u ln |u|, x ∈ RN , where a is a sign-changing function. Many authors pay their attention to find sign-changing solutions to problem (1.2) or similar Kirchhoff-type equations, and some interesting results were obtained. The authors obtained the existence of radial sign-changing solutions with prescribed numbers of nodal domains for Kirchhoff problem (1.6) in Hr1(R3), the subspace of radial functions of H 1(R3) by using a Nehari manifold and gluing solution pieces together, when V (x) = V (|x|), f (x, u) = f (|x|, u) and satisfies some conditions. In [43], the authors obtained the existence of least energy sign-changing solutions of Kirchhoff-type equation with critical growth by using the constraint variational method and the quantitative deformation lemma. For more results on sign-changing solutions for Kirchhoff-type equations, we refer the reader to [16,25,31] and the references therein
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