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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
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Full Text: Euclid