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Existence and asymptotic behavior of boundary blow-up solutions for weighted $$p(x)$$-Laplacian equations with exponential nonlinearities. (English) Zbl 1216.35064
This paper investigates the following $$p(x)$$-Laplacian equations with exponential nonlinearities: $$-\Delta_{p(x)}u+ \rho(x) e^{f(x,u)}=0$$ in $$\Omega$$, $$u(x)\to+\infty$$ as $$d(x,\partial\Omega)\to 0$$, where $$-\Delta_{p(x)}u= -\text{div}(|\nabla u|^{p(x)-2}\nabla u)$$ is called $$p(x)$$-Laplacian, $$\rho(x)\in C(\Omega)$$. The asymptotic behavior of boundary blow-up solutions is discussed, and the existence of boundary blow-up solutions is given.
##### MSC:
 35J70 Degenerate elliptic equations 35B40 Asymptotic behavior of solutions to PDEs 35B44 Blow-up in context of PDEs 35D30 Weak solutions to PDEs
##### Keywords:
$$p(x)$$-Laplacian; blow-up solutions; asymptotic behavior
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##### References:
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