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A two parameters Ambrosetti-Prodi problem. (English) Zbl 0858.35097
The nonlinear boundary value problem $u''+u+f(t,u)= \mu+\nu\varphi(t) \quad\text{in }(0,\pi), \qquad u(0)=u(\pi)=0$ is studied. Here $$f$$ is an $$L^p$$ Carathéodory function, $$\varphi(t)= \sqrt{{2/\pi}}\sin t$$, and $$\mu,\nu\in\mathbb{R}$$. This is a two-parameter version of the well-known Ambrosetti-Prodi problem. One of the main results, Theorem 10, states that if $$\liminf_{|x|\to\infty} f(t,x)=+\infty$$ uniformly in $$t$$ and some additional condition is satisfied, then there exists a nonincreasing Lipschitz function $$\mu_0:\mathbb{R}\to\mathbb{R}$$ such that there is no solution for $$\mu<\mu_0(\nu)$$, at least one solution for $$\mu=\mu_0(\nu)$$ and at least two solutions for $$\mu_0(\nu)<\mu$$. Corollary 11 gives a related result when $$\mu=0$$. A condition implying that $$\mu_0$$ is decreasing is given in Theorem 12 and Corollary 14. The technique of proof involves the use of a notion of strict super and subsolutions stronger than usual, together with degree theory. This extends previous work by Chiappinelli, Mawhin and Nugari and C. Fabry.

##### MSC:
 35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs 34C23 Bifurcation theory for ordinary differential equations 34B15 Nonlinear boundary value problems for ordinary differential equations 35J65 Nonlinear boundary value problems for linear elliptic equations 47H11 Degree theory for nonlinear operators
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