Abstract
In this paper, we prove the existence of infinitely many small solutions to the following quasilinear elliptic equation −Δ p(x) u + |u|p(x)-2 u = f (x, u) in a smooth bounded domain Ω of \({\mathbb{R}^N}\) with nonlinear boundary conditions \({|\nabla u|^{p-2}\frac{\partial u}{\partial\nu} = |u|^{{q(x)-2}}u}\) . We also assume that \({\{q(x) = p^\ast(x)\}\neq \emptyset}\) , where p*(x) = Np(x)/(N − p(x)) is the critical Sobolev exponent for variable exponents. The proof is based on a new version of the symmetric mountain-pass lemma due to Kajikiya, and property of these solutions is also obtained.
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Liang, S., Zhang, J. Infinitely many small solutions for the p(x)-Laplacian operator with nonlinear boundary conditions. Annali di Matematica 192, 1–16 (2013). https://doi.org/10.1007/s10231-011-0209-y
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DOI: https://doi.org/10.1007/s10231-011-0209-y
Keywords
- p(x)-Laplacian
- Generalized Lebesgue-Sobolev spaces
- Nonlinear boundary conditions
- Concentration-compactness principle