k-convex solutions for multiparameter Dirichlet systems with k-Hessian operator and Lane-Emden type nonlinearities

Abstract

Abstract In this article, our main aim is to investigate the existence of radial k k -convex solutions for the following Dirichlet system with k k -Hessian operators: S k ( D 2 u ) = λ 1 ν 1 ( ∣ x ∣ ) ( − u ) p 1 ( − v ) q 1 in ℬ ( R ) , S k ( D 2 v ) = λ 2 ν 2 ( ∣ x ∣ ) ( − u ) p 2 ( − v ) q 2 in ℬ ( R ) , u = v = 0 on ∂ ℬ ( R ) . \left\{\begin{array}{ll}{S}_{k}\left({D}^{2}u)={\lambda }_{1}{\nu }_{1}\left(| x| ){\left(-u)}^{{p}_{1}}{\left(-v)}^{{q}_{1}}& {\rm{in}}\hspace{1em}{\mathcal{ {\mathcal B} }}\left(R),\\ {S}_{k}\left({D}^{2}v)={\lambda }_{2}{\nu }_{2}\left(| x| ){\left(-u)}^{{p}_{2}}{\left(-v)}^{{q}_{2}}& {\rm{in}}\hspace{1em}{\mathcal{ {\mathcal B} }}\left(R),\\ u=v=0& {\rm{on}}\hspace{1em}\partial {\mathcal{ {\mathcal B} }}\left(R).\end{array}\right. Here, u p 1 v q 1 {u}^{{p}_{1}}{v}^{{q}_{1}} is called a Lane-Emden type nonlinearity. The weight functions ν 1 , ν 2 ∈ C ( [ 0 , R ] , [ 0 , ∞ ) ) {\nu }_{1},{\nu }_{2}\in C\left(\left[0,R],\left[0,\infty )) with ν 1 ( r ) > 0 < ν 2 ( r ) {\nu }_{1}\left(r)\gt 0\lt {\nu }_{2}\left(r) for all r ∈ ( 0 , R ] r\in \left(0,R] , p 1 , q 2 {p}_{1},{q}_{2} are nonnegative and q 1 , p 2 {q}_{1},{p}_{2} are positive exponents, ℬ ( R ) = { x ∈ R N : ∣ x ∣ < R } {\mathcal{ {\mathcal B} }}\left(R)=\left\{x\in {{\mathbb{R}}}^{N}:| x| \lt R\right\} , N ≥ 2 N\ge 2 is an integer, N 2 ≤ k ≤ N \frac{N}{2}\le k\le N . In order to achieve our main goal, we first study the existence of radial k k -convex solutions of the above-mentioned systems with general nonlinear terms by using the upper and lower solution method and Leray-Schauder degree. Based on this, by constructing a continuous curve, which divides the first quadrant into two disjoint sets, we obtain the existence and multiplicity of radial k k -convex solutions for the system depending on the parameters λ 1 {\lambda }_{1} , λ 2 {\lambda }_{2} and the continuous curve.