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  • The Nehari manifold approach for singular equations involving the ▫$p(x)$▫-Laplace operator
    Repovš, Dušan, 1954- ; Saoudi, Kamel
    We study the following singular problem involving the ▫$p(x)$▫-Laplace operator ▫$\Delta p(x)u = \mathrm{div}(|\nabla u|^{p(x)-2} \nabla u)$▫, where ▫$p(x)$▫ is a nonconstant continuous function ▫$$ ... (P_\lambda) \quad \begin{cases} -\Delta_{p(x)}u = a(x)|u|^{q(x)-2}u(x) + \frac{\lambda b(x)}{u^{\delta(x)}} & \text{in} \; \Omega, \\ u>0 & \text{in} \; \Omega, \\ u=0 &\text{on} \; \partial\Omega. \end{cases} $$▫ Here, ▫$\Omega$▫ is a bounded domain in ▫$\mathbb{R}^{N \ge 2}$▫ with ▫$C^2$▫-boundary, ▫$\lambda$▫ is a positive parameter, ▫$a(x), b(x) \in C(\overline{\Omega})$▫ are positive weight functions with compact support in ▫$\Omega$▫, and ▫$\delta(x), p(x), q(x) \in C(\overline{\Omega})$▫ satisfy certain hypotheses ▫$(A_0)$▫ and ▫$(A_1)$▫. We apply the Nehari manifold approach and some new techniques to establish the multiplicity of positive solutions for problem ▫$(P_\lambda)$▫.
    Source: Complex variables and elliptic equations. - ISSN 1747-6933 (Vol. 68, no. 1, 2023, str. 135-149)
    Type of material - article, component part ; adult, serious
    Publish date - 2023
    Language - english
    COBISS.SI-ID - 80237571

    Link(s):

    https://www.tandfonline.com/doi/full/10.1080/17476933.2021.1980878
    DOI

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