## 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.

### MSC:

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