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Zbl 1127.35020
Mihăilescu, Mihai; Pucci, Patrizia; Rădulescu, Vicenţiu
Nonhomogeneous boundary value problems in anisotropic Sobolev spaces. (English)
[J] C. R., Math., Acad. Sci. Paris 345, No. 10, 561-566 (2007). ISSN 1631-073X

Summary: We study the nonlinear boundary value problem $$-\sum_{i=1}^N (|u_{x_i}|^{p_i(x)-2} u_{x_i})_{x_i}= \lambda|u|^{q(x)-2} \text{ in }\Omega, \quad u=0\text{ on }\partial\Omega,$$ where $\Omega\subset \Bbb R^N$ $(N\ge 3)$ is a bounded domain with smooth boundary, $\lambda$ is a positive real number, and the continuous functions $p_i$ and $q$ satisfy $2\le p_i(x)<N$ and $q(x)>1$ for any $x\in\overline{\Omega}$ and any $i\in \{1,\dots, N\}$. By analyzing the growth of the functions $p_i$ and $q$ we prove in this note several existence results in Sobolev spaces with variable exponents.

MSC 2000:: *35J65 (Nonlinear) BVP for (non)linear elliptic equations
46E35 Sobolev spaces and generalizations

Keywords: boundary value problems; nonlinear elliptic equations; anisotropic Sobolev spaces

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