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Landesman-Lazer type conditions for a system of \(p\)-Laplacian like operators. (English) Zbl 1119.34010
The authors consider the periodic boundary value problem with a \(\Phi\)-Laplacian operator \[ \Phi(u')'=f(t,u,u'),\;\;\;u(0)=u(T),\tag{1} \] where \(\Phi\) is a (strictly monotone, coercive) homeomorphism of \(\mathbb R^N\) and the vector field \(f\) satisfies some condition of Landesman-Lazer type. A typical result (that may be seen as an extension of a theorem of L. Nirenberg on elliptic systems [see Contrib. Nonlin. Functional Analysis, Proc. Sympos. Univ. Wisconsin, Madison 1971, 1–9 (1971; Zbl 0267.47034)]) is that (1) has at least one solution provided that the following conditions are satisfied: 1) \(f\) is bounded, 2) given bases \(\{e_1,\dots,e_n\}\), \(\{w_1,\dots,w_n\}\subset S^{N-1}\) the limits \[ \limsup_{s\to+\infty}\langle f(t, x+se_i,v),w_i\rangle:=\bar f_i(t) \] and \[ \liminf_{s\to-\infty}\langle f(t, x+se_i,v),w_i\rangle := \underline f_i(t) \] exist uniformly with respect to \(x\in\) span\(\{e_j| \,j\neq i\}\) and \(v\) bounded, 3)\(\int_0^T\bar f_i<0<\int_0^T\underline f_i\), \(i=1,\dots n\).
This theorem holds for unbounded \(f\) if a Nagumo condition is assumed. The main tool of the proof is the homotopy invariance of the Leray-Schauder degree.

34B15 Nonlinear boundary value problems for ordinary differential equations
47J05 Equations involving nonlinear operators (general)
Zbl 0267.47034
Full Text: DOI
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