On the Ambrosetti-Prodi problem for a system involving \(p\)-Laplacian operator (Q2786726)

Under consideration is the problem NEWLINENEWLINE\[NEWLINE -\Delta_p u_1= f_1 (x,u_1, u_2)+t_1 \varphi_1 +h_1,\, -\Delta_p u_2 =f_2 (x,u_1,u_2)+t_2 \varphi_2 +h_2, \eqno{(1)} NEWLINE\]NEWLINENEWLINENEWLINE\[NEWLINEu_1|_\Gamma =u_2|_\Gamma =0, \eqno{(2)}NEWLINE\]NEWLINENEWLINE where \(\Delta_p u=\operatorname{div}(| \nabla u|^{p-2}\nabla u)\), \(x\in G\) (\(G\in\mathbb R^n\) is a bounded domain with smooth boundary \(\Gamma\)), \(t_1\), \(t_2\) are real parameters, and \(\varphi_1\), \(\varphi_2\), \(h_1\), \(h_2\in L_\infty (G)\). The function \(f_i (x,s_1,s_2)\) is nondecreasing in \(s_j\) for any \(i\neq j\), \(i=1,2\). It is also satisfies a natural growth condition in the variables \(s_1\), \(s_2\), and some other additional conditions. It is proven that there exists Lipschitz curves \(\Gamma_1\), \(\Gamma_2\) dividing \(\mathbb R\) into three open sets \(M\), \(N\), \(O\) such that, for \((t_1,t_2)\in M\), for \((t_1,t_2)\in \Gamma_1\cup \Gamma_2\cup O\), and for \((t_1,t_2)\in N\), the problem (1), (2) has respectively at least two solutions, at least one solution, and no solutions.