## Existence and blow-up rate of large solutions of $$p(x)$$-Laplacian equations with large perturbation and gradient terms.(English)Zbl 1375.35144

Summary: We investigate boundary blow-up solutions of the problem $\begin{cases} -\Delta_{p(x)}u+f(x,u)=\rho(x,u)+K(x)|\nabla u|^{m(x)}\;\text{in}\;\Omega,\\ u(x)\rightarrow+\infty\;\text{as}\;d(x,\partial\Omega)\rightarrow 0, \end{cases}$ where $$\Delta_{p(x)}u=\mathrm{div}(|\nabla u|^{p(x)-2}\nabla u)$$ is called $$p(x)$$-Laplacian. Our results extend the previous work of J. García-Melián and A. Suárez [Adv. Nonlinear Stud. 3, No. 2, 193–206 (2003; Zbl 1045.35025)] from the case where $$p(\cdot)\equiv2$$, without gradient term, to the case where $$p(\cdot)$$ is a function, with gradient term. It also extends the previous work of Y. Liang et al. [Taiwanese J. Math. 18, No. 2, 599–632 (2014; Zbl 1357.35144)] from the radial case in the problem to the non-radial case. The existence of boundary blow-up solutions is established and the singularity of boundary blow-up solution is also studied for several cases including when $$\frac{\rho(x,u(x))}{f(x,u(x))}\rightarrow 1$$ as $$x\rightarrow\partial\Omega$$, which means that $$\rho(x,u)$$ is a large perturbation. We provide an exact estimate of the pointwise different behavior of the solutions near the boundary in terms of $$d(x,\partial\Omega)$$. Hence, the results of this paper are new even in the canonical case $$p(\cdot)\equiv2$$. In particular, we do not have the comparison principle, because we don’t make the monotone assumption of nonlinear term.

### MSC:

 35J25 Boundary value problems for second-order elliptic equations 35B40 Asymptotic behavior of solutions to PDEs 35B44 Blow-up in context of PDEs 35J60 Nonlinear elliptic equations

### Citations:

Zbl 1045.35025; Zbl 1357.35144
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