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On the solvability of singular boundary value problems on the real line in the critical growth case

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  • Combining fixed point techniques with the method of lower-upper solutions we prove the existence of at least one weak solution for the following boundary value problem

    $ \begin{equation*} \left\{ \begin{array}{ll} \left( \, \Phi(a(t, x(t)) \, x'(t) )\, \right)' = f(t, x(t), x'(t)) &\mbox{ in } \mathbb{R}\\ x(-\infty) = \nu_{1}, \quad x(+\infty) = \nu_{2} \end{array} \right. \end{equation*} $

    where $ \nu_{1}, \nu_{2}\in \mathbb{R} $, $ \Phi: \mathbb{R} \rightarrow \mathbb{R} $ is a strictly increasing homeomorphism extending the classical $ p $-Laplacian, $ a $ is a nonnegative continuous function on $ \mathbb{R} \times \mathbb{R} $ which can vanish on a set having zero Lebesgue measure and $ f $ is a Carathéodory function on $ \mathbb{R} \times \mathbb{R}^{2} $.

    Mathematics Subject Classification: Primary: 34B40, 34C37; Secondary: 34B16, 34L30.


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