On regions of existence and nonexistence of solutions for a system of \(p-q\)-Laplacians. (English) Zbl 1157.35350

Clément, Philippe; García-Huidobro, Marta; Guerra, Ignacio; Manásevich, Raúl

Asymptotic Anal. 48, No. 1-2, 1-18 (2006).

Summary: We give a new region of existence of solutions to the superhomogeneous Dirichlet problem \[ \begin{aligned} -\Delta_{p}u&=v^{\delta},\quad v>0 \,{\text{in}}\, B,\\-\Delta_{q}v&=u^{\mu},\quad u>0\, {\text{in}}\, B,\\u=v&=0 \quad {\text{on}}\, \partial B \end{aligned} \] where \(B\) is the ball of radius \(R>0\) centered at the origin in \(\mathbb{R}^N\). Here \(\delta, \mu >0\) and \(\Delta_{m} u = {\text{div}}(|\nabla u|^{m-2} \nabla u)\) is the \(m\)-Laplacian operator for \(m>1\).

MSC:

35J55	Systems of elliptic equations, boundary value problems (MSC2000)
35J60	Nonlinear elliptic equations

Keywords:

\(m\)-Laplacian; energy identities