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

This paper investigates the existence, multiplicity, and nonexistence of symmetric positive solutions for the fourth-order n-dimensional m-Laplace system { ϕ m ( x ″ ( t ) ) ) ″ = Ψ ( t ) f ( t , x ( t ) ) , 0 < t < 1 , x ( 0 ) = x ( 1 ) = ∫ 0 1 g ( s ) x ( s ) d s , ϕ m ( x ″ ( 0 ) ) = ϕ m ( x ″ ( 1 ) ) = ∫ 0 1 h ( s ) ϕ m ( x ″ ( s ) ) d s . $$\left \{ \textstyle\begin{array}{l} \phi_{m}(\mathbf{x}{''}(t))){''}=\Psi(t)\mathbf{f}(t,\mathbf{x}(t)), \quad 0< t< 1,\\ \mathbf{x}(0)=\mathbf{x}(1)=\int_{0}^{1}\mathbf{g}(s)\mathbf{x}(s)\, ds,\\ \phi_{m}(\mathbf{x}{''}(0))=\phi_{m}(\mathbf{x}{''}(1))=\int _{0}^{1}\mathbf{h}(s)\phi_{m}(\mathbf{x}{''}(s))\,ds. \end{array}\displaystyle \right . $$ The vector-valued function x is defined by x = [ x 1 , x 2 , … , x n ] ⊤ $\mathbf {x}=[x_{1},x_{2},\dots,x_{n}]^{\top}$ , Ψ ( t ) = diag [ ψ 1 ( t ) , … , ψ i ( t ) , … , ψ n ( t ) ] $\Psi(t)=\operatorname{diag}[\psi_{1}(t), \ldots, \psi _{i}(t), \ldots, \psi_{n}(t)]$ , where ψ i ∈ L p [ 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.