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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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References:

[1] Bechah A., Electron. J. Differential Equations 2001 pp 1–
[2] Berger M., Lectures on Nonlinear Problems in Mathematical Analysis, in: Nonlinearity and Functional Analysis (1975)
[3] DOI: 10.1016/0362-546X(86)90011-8 · Zbl 0593.35045
[4] DOI: 10.1007/s00030-002-8130-0 · Zbl 1011.35050
[5] DOI: 10.1016/S0022-0396(02)00112-2 · Zbl 1021.35034
[6] Chabrowski J., Rev. Math. Univ. Complut. Madrid 9 pp 207–
[7] DOI: 10.1155/S1085337502000829 · Zbl 1005.35036
[8] de Thélin F., C. R. Acad. Sci. Paris Ser. I 321 pp 589–
[9] de Thélin F., Rev. Mat. Apl. 13 pp 1–
[10] de Thélin F., Rev. Math. Univ. Complut. Madrid 6 pp 153–
[11] Diaz J. I., C. R. Acad. Sci. Paris 305 pp 521–
[12] Kandilakis D. A., Electron. J. Differential Equations 2005 pp 1–
[13] Drabek P., Differential Integral Equations 16 pp 1519– · Zbl 0425.34042
[14] El-Zahrani E. A., Electron. J. Differential Equations 2006 pp 1–
[15] Felmer P., Comm. Partial Differential Equations 17 pp 2013–
[16] DOI: 10.1155/S1085337504403078 · Zbl 1129.35372
[17] Fernandez-Bonder J., J. Differential Equations 245 pp 845–
[18] Zhang G., Electron. J. Differential Equations 2005 pp 1–
[19] DOI: 10.1007/BF01449041 · Zbl 0561.35003
[20] DOI: 10.1016/S0362-546X(02)00151-7 · Zbl 1087.35043
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