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Nonhomogeneous boundary value problems for some nonlinear equations with singular \(\phi\)-Laplacian. (English) Zbl 1170.34014
The paper studies existence and multiplicity results for the quasilinear equation \[ (\phi(u'))'=f(t,u,u') \] under nonhomogeneous Dirichlet conditions
\[ u(0)=A,\qquad u(T)=B \] and Neumann-Steklov boundary conditions
\[ \phi(u'(0)) = g_0(u(0)),\qquad \phi(u'(T)) = g_T(u(T)). \] It is assumed that \(\phi:(-a,a)\to \mathbb{R}\) is an increasing homeomorphism such that \(\phi(0)=0\), and \(f\), \(g_0\) and \(g_T\) are continuous functions. Using the well-known Leray-Schauder approach, it is proved that the problem has a solution for any right-hand term \(f\), provided that the Dirichlet boundary data satisfies the (sharp) condition \(|B-A|<aT\). Moreover, the Neumann-Steklov problem is solved under suitable sign conditions of Villari’s type. Furthermore, the method of upper and lower solutions is extended to the case of nonhomogeneous Neumann conditions, without restrictions of Nagumo type. The combination of this method with Leray-Schauder degree allows the author to prove an Ambrosetti-Prodi type multiplicity result.

MSC:
34B15 Nonlinear boundary value problems for ordinary differential equations
47N20 Applications of operator theory to differential and integral equations
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