Symmetric positive solutions for fourth-order n-dimensional m-Laplace systems

Abstract

Abstract This paper investigates the existence, multiplicity, and nonexistence of symmetric positive solutions for the fourth-order n-dimensional m-Laplace system {ϕm(x″(t)))″=Ψ(t)f(t,x(t)),0<t<1,x(0)=x(1)=∫01g(s)x(s)ds,ϕm(x″(0))=ϕm(x″(1))=∫01h(s)ϕm(x″(s))ds. $$\left \{ \textstyle\begin{array}{l} \phi_{m}(\mathbf{x}{''}(t))){''}=\Psi(t)\mathbf{f}(t,\mathbf{x}(t)), \quad 0< t< 1,\\ \mathbf{x}(0)=\mathbf{x}(1)=\int_{0}^{1}\mathbf{g}(s)\mathbf{x}(s)\, ds,\\ \phi_{m}(\mathbf{x}{''}(0))=\phi_{m}(\mathbf{x}{''}(1))=\int _{0}^{1}\mathbf{h}(s)\phi_{m}(\mathbf{x}{''}(s))\,ds. \end{array}\displaystyle \right . $$ The vector-valued function x is defined by x=[x1,x2,…,xn]⊤ $\mathbf {x}=[x_{1},x_{2},\dots,x_{n}]^{\top}$, Ψ(t)=diag[ψ1(t),…,ψi(t),…,ψn(t)] $\Psi(t)=\operatorname{diag}[\psi_{1}(t), \ldots, \psi _{i}(t), \ldots, \psi_{n}(t)]$, where ψi∈Lp[0,1] $\psi_{i}\in L^{p}[0,1]$ for some p≥1 $p\geq1$. Our methods employ the fixed point theorem in a cone and the inequality technique. Finally, an example illustrates our main results.

Keywords

• Symmetric positive solutions
• n-dimensional system
• m-Laplace operator
• Matrix theory
• Fixed point technique