Blow-up rates of large solutions for a \(\phi\)-Laplacian problem with gradient term (Q2877941)

Let \(\Omega\) be a domain in \(\mathbb{R}^N\) satisfying a uniform internal (external) sphere condition. The authors consider a \(\phi\)-Laplacian problem of the type NEWLINE\[NEWLINE\text{div}\biggl( \phi(|\nabla u|)\frac{\nabla u}{|\nabla u|}\biggr)+g(|\nabla u|)=f(u) \;\;\text{in} \;\;\Omega, \;\;\;u=+\infty \;\;\text{on} \;\;\partial \OmegaNEWLINE\]NEWLINE \noindent and study the exact asymptotic behavior of the solutions near the boundary. Here, \(\phi,f,g\) are increasing differentiable real functions which satisfy the following conditions: \(\phi\) is an odd homeomorphism, \(f\) and \(f'\) are positive in \(\mathbb{R}_+\), \(g(0)=0\), \(f(t),g(t)\rightarrow +\infty\) as \(t\rightarrow +\infty\), and there exist \(\rho,\gamma,\tau>-1\) such that \(\phi',g'\) are regularly varying at infinity with index \(\rho\) and \(\gamma\), respectively, and \(\phi'\) is regularly varying at \(0\) with index \(\tau\).NEWLINENEWLINE\noindent Let \(\displaystyle{F(s)=\int_0^sf(t)dt, \;\;H(s)=\int_0^{\phi(s)}\phi^{-1}(r)dr}\), \ \ for all \(s\in [0,+\infty)\),NEWLINENEWLINE\noindent and let \(\alpha=H\circ g^{-1}\circ f\).NEWLINENEWLINENEWLINEThe main result of this paper states that under the following conditionsNEWLINENEWLINE\noindent \(1)\) \ \(\frac{\alpha(s)}{s}\) is increasing in \((\bar{s},+\infty)\) for some \(\bar{s}>0\), and \(\displaystyle{\lim_{s\rightarrow +\infty}}\) \(\frac{\alpha(s)}{s}=+\infty\);NEWLINENEWLINE\noindent \(2)\) \ \(\Psi_1(t):=\int_t^{+\infty}\frac{ds}{H^{-1}((F(s))}<+\infty\) for \(t\in \mathbb{R}_+\), \(\displaystyle{\liminf_{t\rightarrow +\infty}}\) \(\frac{\Psi_1(\beta t)}{\Psi_1(t)}>1\) \ \ for \(\beta\in (0,1)\), \ and \ \(\displaystyle{\lim_{t\rightarrow +\infty}}\) \(\frac{g(H^{-1}(F(t)))}{f(t)}=0\)NEWLINENEWLINE(resp. \(\Psi_2(t):=\int_t^{+\infty}\frac{ds}{g^{-1}((f(s))}<+\infty\) for \(t\in \mathbb{R}_+\), \ \(\displaystyle{\liminf_{t\rightarrow +\infty}}\) \(\frac{\Psi_2(\beta t)}{\Psi_2(t)}>1\) \ \ for \(\beta\in (0,1)\), \ and \ \(\displaystyle{\lim_{t\rightarrow +\infty}}\) \(\frac{\alpha'(t)}{f(t)}=0\)),NEWLINENEWLINE\noindent any solution \(u\) of the above problem has the following behaviorNEWLINENEWLINE\(\displaystyle{\frac{u(x)}{\Psi_1^{-1}(d(x,\partial \Omega))}\rightarrow 1}\) (resp. \(\displaystyle{\frac{u(x)}{\Psi_2^{-1}(d(x,\partial \Omega))}\rightarrow 1}\)) as \(x\rightarrow \partial \Omega\).NEWLINENEWLINE\noindent The authors first study the problem in the one dimensional case and, successively, for the case of radial solutions in balls and annuli. Then, from the results for these cases, the authors derive the main result using a comparison principle for couples of solutions of the above problem in domains \(\Omega_1,\Omega_2\) with \(\Omega_1\subset \Omega_2\).