Abstract
In this paper, we consider the following p-Kirchhoff equation: (P)−[M(∥u∥p)]p−1Δpu=f(x,u)in 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}$$(\\mathrm{P})\\quad-\\bigl[M\\bigl(\\Vert u\\Vert ^{p}\\bigr) \\bigr]^{p-1}\\Delta_{p} u=f(x,u)\\quad\\mbox{in } \\Omega $$\\end{document} with Dirichlet boundary conditions, where Ω is a bounded domain in mathbb{R}^{N}. Under proper assumptions on M and f, we obtain three existence theorems of infinitely many solutions for problem (P) by the fountain theorem. Moreover, for a special nonlinearity f(x,u)=lambda |u|^{q-2}u+|u|^{r-2}u (1< q< p< r< p^{*}), we prove that problem (P) has at least two nonnegative solutions via the Nehari manifold method and a sequence of solutions with negative energy by the dual fountain theorem.
Highlights
In this paper, we consider the following p-Kirchhoff equation:– M u p p– pu = f (x, u) in, u = on ∂, ( . )where M, f are continuous functions, is a bounded domain in RN with smooth boundary, u p = |∇u|p dx ( < p < N )
Much attention has been paid to the existence of nontrivial solutions, sign-changing solutions, ground state solutions, multiplicity of solutions and concentration of solutions for the following case:
Using the Nehari manifold and fibering maps, they established the existence of multiple positive solutions for ( . )
Summary
We consider the following p-Kirchhoff equation:. where M, f are continuous functions, is a bounded domain in RN with smooth boundary, u p = |∇u|p dx ( < p < N ). Huang et al Boundary Value Problems (2017) 2017:41 mountain pass theorem. Using the Nehari manifold and fibering maps, they established the existence of multiple positive solutions for Alves et al [ ] and Corrêa and Figueiredo [ ] showed that the problem has a positive solution by the mountain pass theorem, where M is supposed to satisfy the following conditions:. In [ ], Liu established the existence of infinite solutions to a Kirchhoff-type equation like ). By the fountain theorem and dual fountain theorem, they investigated the problem with M satisfying (M ) and (M ) M(t) ≤ m for all t >. Motivated mainly by [ , , ], we shall establish conditions on M and f under which problem We suppose that f satisfies the following conditions:
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