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  • The variational analysis of a nonlinear anisotropic problem with no-flux boundary condition
    Afrouzi, Ghasem A. ; Mirzapour, Maryam ; Rǎdulescu, Vicenţiu, 1958-
    We study the nonlinear degenerate anisotropic problem ▫$$\begin{cases} -\sum_{i=1}^N \partial_{x_i}a_i (x, \partial_{x_i}u) + b(x)|u|^{p_M(x)-2}u = \lambda |u|^{q(x)-2}u & \text{in} \quad \Omega,\\ ... u(x) = \text{constant} & \text{on} \quad \partial \, \Omega\\ \sum_{i=1}^N \int_{\partial\Omega} a_i (x, \partial_{x_i}u)v_ido =0, & \end{cases}$$▫ where ▫$\Omega \subset \mathbb{R}^N$▫ is a bounded domain with smooth boundary. The constant value of the boundary data is not specified, whereas the zero integral term corresponds to a no-flux boundary condition. In the case when ▫$|u|^{q(x)-2}u$▫ "dominates" the left-hand side, we show that a nontrivial solution exists for all positive values of ▫$\lambda$▫. If the term ▫$|u|^{q(x)-2}u$▫ is dominated by the left-hand side, we prove that a solution exists either for small or for large values of ▫$\lambda > 0$▫. The proofs combine variational arguments with energy estimates.
    Source: Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A, Matemáticas. - ISSN 1578-7303 (Vol. 109, no. 2, 2015, str. 581-595)
    Type of material - article, component part ; adult, serious
    Publish date - 2015
    Language - english
    COBISS.SI-ID - 17439833

    Link(s):

    http://dx.doi.org/10.1007/s13398-014-0202-6

    Omejen dostop



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