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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.

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