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A nonlinear eigenvalue problem with $ p(x) $-growth and generalized Robin boundary value condition

  • * Corresponding author: Somayeh Saiedinezhad

    * Corresponding author: Somayeh Saiedinezhad
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  • We are concerned with the study of the following nonlinear eigenvalue problem with Robin boundary condition

    $\begin{cases} -{\rm div}\,(a(x,\nabla u))=λ b(x,u)&\mbox{in} \ Ω\\\dfrac{\partial A}{\partial n}+β(x) c(x,u)=0&\mbox{on}\\partialΩ.\end{cases}$

    The abstract setting involves Sobolev spaces with variable exponent. The main result of the present paper establishes a sufficient condition for the existence of an unbounded sequence of eigenvalues. Our arguments strongly rely on the Lusternik-Schnirelmann principle. Finally, we focus to the following particular case, which is a $p(x)$-Laplacian problem with several variable exponents:

    $\begin{cases} -{\rm div}\,(a_0(x) |\nabla u|^{p(x)-2}\nabla u)=λ b_0(x)|u|^{q(x)-2}u&\mbox{in} \ Ω\\|\nabla u|^{p(x)-2}\dfrac{\partial u}{\partial n}+β(x)|u|^{r(x)-2} u=0&\mbox{on}\\partialΩ.\end{cases}$

    Combining variational arguments, we establish several properties of the eigenvalues family of this nonhomogeneous Robin problem.

    Mathematics Subject Classification: Primary:34B09, 35P30;Secondary:35J20.


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