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Zbl 1180.35242
Manfredi, J.J.; Rossi, J.D.; Urbano, J.M.
$p(x)$-harmonic functions with unbounded exponent in a subdomain. (English)
[J] Ann. Inst. Henri Poincaré, Anal. Non Linéaire 26, No. 6, 2581-2595 (2009). ISSN 0294-1449

Summary: We study the Dirichlet problem $-\text{div}(|\nabla u|^{p(x)-2}\nabla u)=0$ in $\Omega$, with $u=f$ on $\partial\Omega$ and $p(x)=\infty$ in $D$, where $D$ is a subdomain of the reference domain $\Omega $. The main issue is to give a proper sense to what a solution is. To this end, we consider the limit as $n\rightarrow \infty $ of the solutions $u_n$ to the corresponding problem when $p_n(x)=p(x)\land n$, in particular, with $p_n=n$ in $D$. Under suitable assumptions on the data, we find that such a limit exists and that it can be characterized as the unique solution of a variational minimization problem which is, in addition, $\infty $-harmonic within $D$. Moreover, we examine this limit in the viscosity sense and find the boundary value problem it satisfies in the whole $\Omega$.

MSC 2000:: *35J62
35J25 Second order elliptic equations, boundary value problems
35J70 Elliptic equations of degenerate type
35J92

Keywords: $p(x)$-Laplacian; infinity-Laplacian; viscosity solutions

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Zentralblatt MATH Berlin [Germany]