Nonzero solutions of a nonlinear elliptic system at resonance. (English) Zbl 0921.35051

The existence of nonzero solutions of a nonlinear elliptic system at resonance: \[ -\Delta u=\lambda u+\delta v+g_1(u),\quad -\Delta v= \theta u+\gamma v+g_2(v)\tag{s} \] is studied. This problem (s) can be treated by using a decoupling technique, but this approach is difficult to apply to systems with three or more equations. The author rewrites this problem as: \[ -\Delta U=A (U)+G(U)\tag{p} \] and then proves that problem (p) has at least a nonzero solution in several cases by the approach of the fixed point theory. It is pointed out by the author that the result obtained here can be also applied to a class of nonlinear biharmonic equations under Navier and Dirichlet conditions.


35J55 Systems of elliptic equations, boundary value problems (MSC2000)
35J60 Nonlinear elliptic equations


Brouwer degree
Full Text: DOI


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