Bifurcation and multiplicity results for classes of \(p, q\)-Laplacian systems. (English) Zbl 1376.35084

Shivaji, Ratnasingham; Son, Byungjae

Topol. Methods Nonlinear Anal. 48, No. 1, 103-114 (2016).

The authors are interested in the study of solutions of the following boundary value problems \[ \begin{cases} -\Delta_p u = \lambda \{u^{p-1-\alpha}+f(v)\}&\text{in}\;\Omega, \\ -\Delta_q v = \lambda \{v^{q-1-\beta}+g(u)\}&\text{in}\;\Omega,\\ u=v=0 &\text{on}\;\partial\Omega,\end{cases}{(1)} \] where \(\Delta_m u\equiv \mathrm{div }(|\nabla u|^{m-2}u)\), \(m>1 \), is the \(m\)-Laplacian operator, \(\lambda >0\), \(p, q>1\), \(\alpha \in (0, p-1)\), \(\beta \in (0, q-1) \) are parameters and \(\Omega\) is a bounded domain in \(\mathbb R^N\), \(N\geq 1\), with smooth boundary \(\partial\Omega\). Here \(f, g: [0, \infty)\rightarrow \mathbb R\) are nondecreasing continuous functions with \(f(0)=g(0)=0\).
Under some additional conditions on \(f\) and \(g\) and using the method of sub-super solutions, the authors obtain bifurcation and multiplicity results to problem (1).

Reviewer: Said El Manouni (Berlin)

MSC:

35J92	Quasilinear elliptic equations with \(p\)-Laplacian
35J40	Boundary value problems for higher-order elliptic equations

Keywords:

\(p\)-Laplacian; quasilinear systems; bifurcation; multiplicity results