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

Abstract

In this article, our main aim is to investigate the existence of radial kk-convex solutions for the following Dirichlet system with kk-Hessian operators: Sk(D2u)=λ1ν1(∣x∣)(−u)p1(−v)q1inℬ(R),Sk(D2v)=λ2ν2(∣x∣)(−u)p2(−v)q2inℬ(R),u=v=0on∂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, up1vq1{u}^{{p}_{1}}{v}^{{q}_{1}} is called a Lane-Emden type nonlinearity. The weight functions ν1,ν2∈C([0,R],[0,∞)){\nu }_{1},{\nu }_{2}\in C\left(\left[0,R],\left[0,\infty )) with ν1(r)>0<ν2(r){\nu }_{1}\left(r)\gt 0\lt {\nu }_{2}\left(r) for all r∈(0,R]r\in \left(0,R], p1,q2{p}_{1},{q}_{2} are nonnegative and q1,p2{q}_{1},{p}_{2} are positive exponents, ℬ(R)={x∈RN:∣x∣<R}{\mathcal{ {\mathcal B} }}\left(R)=\left\{x\in {{\mathbb{R}}}^{N}:| x| \lt R\right\}, N≥2N\ge 2 is an integer, N2≤k≤N\frac{N}{2}\le k\le N. In order to achieve our main goal, we first study the existence of radial kk-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 kk-convex solutions for the system depending on the parameters λ1{\lambda }_{1}, λ2{\lambda }_{2} and the continuous curve.

Keywords

• k-hessian operator
• lane-emden nonlinearity
• k-convex solutions
• existence
• leray-schauder degree
• lower and upper solution method
• 34b15
• 34b18
• 35j47
• 35j96