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Multiple solutions for degenerate elliptic systems near resonance at higher eigenvalues. (English) Zbl 1250.35092

Summary: We study the degenerate semilinear elliptic systems of the form \(-\text{div}(h_1(x)\nabla u) = \lambda(a(x)u + b(x)v) + F_u(x, u, v), x \in \Omega, -\text{div}(h_2(x)\nabla v) = \lambda(d(x)v + b(x)u) + F_v(x, u, v), x \in \Omega, u|_{\partial \Omega} = v|_{\partial \Omega} = 0\), where \(\Omega \subset R^N(N \geq 2)\) is an open bounded domain with smooth boundary \(\partial \Omega\), the measurable, nonnegative diffusion coefficients \(h_1, h_2\) are allowed to vanish in \(\Omega\) (as well as at the boundary \(\partial \Omega\)) and/or to blow up in \(\overline{\Omega}\). Some multiplicity results of solutions are obtained for the degenerate elliptic systems which are near resonance at higher eigenvalues by the classical saddle point theorem and a local saddle point theorem in critical point theory.

MSC:

35J48 Higher-order elliptic systems
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