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<p>The first main result of this paper is inspired in the pioneer works (Crandall and Rabinowitz in J Funct Anal 8:321–340, 1971, <a href="https://doi.org/10.1016/0022-1236(71)90015-2">https://doi.org/10.1016/0022-1236(71)90015-2</a>; Rabinowitz in J. Funct Anal 7:487–513, 1971, <a href="https://doi.org/10.1016/0022-1236(71)90030-9">https://doi.org/10.1016/0022-1236(71)90030-9</a>; Rabinowitz in: Zarantonello, Proceedings of a Symposium at the University of Wisconsin, Madison, Wis., 1971) and principally devoted to complementing both Theorem 3.5 of (Rabinowitz 1971) and Theorem 2.2 of (Arcoya et al. in J Funct Anal 268:2298–2335, 2015, <a href="https://doi.org/10.1016/j.jfa.2015.01.014">https://doi.org/10.1016/j.jfa.2015.01.014</a>). It complements Theorem 3.5 of (Rabinowitz 1971), principally, by showing existence of a connected set of solutions for problems that not necessarily have a priori boundedness of solutions. Additionally, it provides additional insight to Theorem 2.2 of (Arcoya et al. 2015) in at least two directions: first by requiring just that the solution from which emanates a connected set of solutions should be isolated instead of unique, and second by assuming that the operator is defined just on open subsets of the parameter-working space. We apply this new result to establish existence of connected branches of strongly-positive classical solutions for Dirichlet problems headed by quasilinear Schrödinger (Theorem <InternalRef RefID="FPar3">2</InternalRef>) and Carrier-type (Theorem <InternalRef RefID="FPar5">3</InternalRef>) operators. A fine qualitative study about these connected branches is presented as well.</p>

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Connected set of solutions from a continuation theorem on open sets

  • 期刊论文

  • 发表日期：

    2025年8月16日

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  • Carlos Alberto Santos,
  • Willian Cintra,
  • Vinicius Ramos

摘要

The first main result of this paper is inspired in the pioneer works (Crandall and Rabinowitz in J Funct Anal 8:321–340, 1971, https://doi.org/10.1016/0022-1236(71)90015-2; Rabinowitz in J. Funct Anal 7:487–513, 1971, https://doi.org/10.1016/0022-1236(71)90030-9; Rabinowitz in: Zarantonello, Proceedings of a Symposium at the University of Wisconsin, Madison, Wis., 1971) and principally devoted to complementing both Theorem 3.5 of (Rabinowitz 1971) and Theorem 2.2 of (Arcoya et al. in J Funct Anal 268:2298–2335, 2015, https://doi.org/10.1016/j.jfa.2015.01.014). It complements Theorem 3.5 of (Rabinowitz 1971), principally, by showing existence of a connected set of solutions for problems that not necessarily have a priori boundedness of solutions. Additionally, it provides additional insight to Theorem 2.2 of (Arcoya et al. 2015) in at least two directions: first by requiring just that the solution from which emanates a connected set of solutions should be isolated instead of unique, and second by assuming that the operator is defined just on open subsets of the parameter-working space. We apply this new result to establish existence of connected branches of strongly-positive classical solutions for Dirichlet problems headed by quasilinear Schrödinger (Theorem 2) and Carrier-type (Theorem 3) operators. A fine qualitative study about these connected branches is presented as well.