A criterion for existence of a positive solution of a nonlinear elliptic system. (English) Zbl 1161.35012

The author treats an elliptic system with gradient structure, governed by the pair \((p,q)\)-Laplacian, posed on a smoothly bounded domain \(\Omega\) in \(\mathbb{R}^N\): \[ \begin{cases} -\Delta_{p_1}u=\frac{\partial H}{\partial u}(x;u,v), &x\in\Omega,\\ -\Delta_{p_2}v=\frac{\partial H}{\partial v}(x;u,v), &x\in\Omega,\\ u(x)=v(x)=0, &x\in\partial\Omega. \end{cases}\tag{1} \] Here \(1<p_i<N\) and \(\Delta_p := \text{div} (| \nabla| ^{p-2}\nabla)\) denotes the \(p\)-Laplacian, as usual. The potential function \(H\) is given by \[ H(x;s,t):=\int_0^sh_1(x;r)\,dr+\int_0^th_2(x;r)\,dr +\left(\int_0^sg_1(x;r)\,dr\right) \left(\int_0^tg_2(x;r)\,dr\right), \] where \(h_i\) and \(g_i\) are positive Caratheodory functions, \(i=1,2\). It is assumed that the functions \(h_i\) are asymptotically homogeneous of orders \(p_i-1\) in the second argument \(r\), for \(r\) near 0 and \(\infty\), and that the functions \(g_i\) are asymptotically homogeneous near \(r=0\), of orders less than \(p_i-1\). Some monotonicity conditions are imposed on \(h_i\) and \(g_i\), and concavity conditions on \(g_i\).
The result states a sharp criterion for the existence of a unique positive solution to (1). It is formulated as a sign condition on the lowest eigenvalues of three related nonlinear eigenvalue problems. The proof is variational and employs the gradient structure of the problem.


35J55 Systems of elliptic equations, boundary value problems (MSC2000)
35J50 Variational methods for elliptic systems
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