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

Velin, J.

Anal. Appl., Singap. 6, No. 3, 299-321 (2008).

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.

Reviewer: Nils Ackermann (México)

MSC:

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

Keywords:

elliptic system; \(p\)-Laplacian; positive solution; nonlinear eigenvalue

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