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Nonuniform nonresonance at the first eigenvalue of the \(p\)-Laplacian. (English) Zbl 0876.35039
The author considers the Dirichlet boundary value problem on the interval \([0,\omega]\) for the equation \[ (\phi_p(u'(t)))'= f(t,u(t),u'(t)),\tag{*} \] where \(u(t)=(u_1(t),...,u_m(t))\), \(\phi_p(u_1,...,u_m)=(|u_1|^{p-2}u_1,...,|u_m|^{p-2}u_m)\), \(1<p<\infty\), \(f\) is an \(L^1\)-Caratheodory mapping. He also studies the associated eigenvalue problem, \(-(\phi_p(u'))'=\lambda\phi_p(u)\), \(u(0)=u(\omega)=0\). Under some additional conditions on \(f\), involving the first eigenvalue \(\lambda_1\), it is proved that the equation (*) has at least one solution \(u\in C^1\) such that \(\phi_p(u')\) is absolutely continuous.

MSC:
35J65 Nonlinear boundary value problems for linear elliptic equations
34B15 Nonlinear boundary value problems for ordinary differential equations
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