Abstract
In this paper, we are concerned with an elliptic equation defined on \( {\mathbb {R}}^N,\ N \ge 1, \) and involving the \( p(u)-\)Laplacian. When \( p(u) = p(u(x)),\ x \in {\mathbb {R}}^N, \) i.e., when p depends on the variable \( x \in {\mathbb {R}}^N \) (through the unknown solution u), we say that we are dealing with the local case of the problem. In this case the \( p(u)-\)Laplacian can be considered as a new class of operators with variable exponents. When \( p(u) = p( \alpha (u)) \) where \( \alpha \) is a scalar function of the unknown solution u, we say that we are dealing with the nonlocal case of the problem. In the present work, the issue of the existence of nontrivial solution in the both cases is addressed.
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Communicated by Rosihan M. Ali.
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Aouaoui, S. An Existence Result to Some Local and Nonlocal \( p(u)-\)Laplacian Problem Defined on \( {\mathbb {R}}^N \). Bull. Malays. Math. Sci. Soc. 46, 123 (2023). https://doi.org/10.1007/s40840-023-01516-z
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DOI: https://doi.org/10.1007/s40840-023-01516-z