Existence and asymptotic behavior of blow-up solutions to weighted quasilinear equations
Let \(\Omega\) be a bounded domain in \(\mathbb R^N\), let \(f: [0, R)\to [0, R)\) be a non decreasing continuous function with \(f(0)=0\), and let \(g: \Omega \to [0, R)\) be a continuous function. The author investigates local weak solutions to the equation \(\Delta_pu=g(x)f(u)\) in \(\Omega\) such that \(u(x)\to +\infty\) as \(x\to\partial\Omega\), where \(p>1\) and \(\Delta_p\) denotes the usual \(p\)-Laplacian. For \(p=2\) and \(g(x)\) bounded and strictly positive, this boundary blow-up problem has a long history and is now well understood. Even the case \(p>1\) has been considered for particular weights \(g\). In the present paper \(g(x)\) satisfies a quite mild condition which allows it to be unbounded near the boundary and to vanish in large parts of \(\Omega\). For this class of functions \(g\), it is shown that if there exists \(w\in W^{1,p}_0(\Omega)\) such that \(\Delta_p w=-g(x)\) in the weak sense and if \(f\) satisfies the growth condition \(\int^\infty (\int_0^tf(\tau)\,d\tau)^{-1/p} \,dt<\infty \) then the problem in above admits a non negative local weak blow-up solution \(u\in W^{1,p}_{loc}(\Omega)\cap C(\Omega)\). Furthermore, assuming a condition on the growth of \(g\) near \(\partial\Omega\), a lower bound of \(u(x)\) in terms of the distance of \(x\) from \(\partial\Omega\) is found; assuming \(g(x)>0\) near \(\partial\Omega\), an asymptotic behaviour of \(u(x)\) near the boundary is given. Finally, examples of functions \(g\) for which the above blow-up problem admits or fails to admit solutions are given.
scientific article; zbMATH DE number 2107288
scientific article