On the existence and asymptotic behavior of large solutions for a semilinear elliptic problem in \(\mathbb R^N\)
The author is concerned with existence and asymptotic behavior of positive large solutions for the problem \(\Delta u=p(x)f(u)\) in \(\mathbb{R}^N\) \((N\geq 3)\) and \(u(x)\rightarrow\infty\) as \(| x| \rightarrow \infty\). It is assumed that \(f\in C^1(0,\infty)\cap C[0,\infty)\) satisfies \(f(0)=0\), \(f'\geq 0\), \(f>0\) on \((0,\infty)\) and \(\int_1^\infty 1/f(t)\,dt=\infty\). Also \(p\) is a smooth nonnegative and continuous function such that \(\int_0^\infty r\min_{| x| =r} p(x)\,dr=\infty\). The first result of the paper establishes the existence of infinitely many large solutions under the assumption \[ \int_0^\infty r(\max_{| x| =r}p(x)-\min_{| x| =r}p(x))(f\circ\Gamma)(r)\,dr<\infty, \] where \(\Gamma=G^{-1}\Big(2/(N-2)\int_0^r s\max_{| x| =s}p(x)\,ds\Big)\) and \(G^{-1}\) is the inverse of \(t\mapsto \int_1^t 1/f(s)\,ds\). The second part of the paper is concerned with the asymptotic behavior of large solutions. It is assumed that \(c_1u^\gamma\leq f(u)\leq c_2 u^\gamma\) for all \(u\geq 0\) and some \(c_1,c_2, 0<\gamma<1\) and \(p\) is a mild oscillation of a radial function \(p_0(| x| )\) in the sense that \(p(x)=p_0(| x| )+O(| x| ^{-\sigma})\) with \(c_3 | x| ^{-k}\leq p_0(| x| )\leq c_4 | x| ^{-k}\), \(k\leq 2\), \(\sigma>k+(2-k)/(1-\gamma)\). Under these hypotheses it is proved that there exist infinitely many positive large solutions \(u\) such that \(u(x)\) behaves at infinity like \(| x| ^{(2-k)/(1-\gamma)}\) if \(k<2\) and like \(\ln^{1/(1-\gamma)}| x| \) if \(k=2\). The proofs rely on constructing suitable upper and lower solutions combined with various comparison principles.
scientific article; zbMATH DE number 2177448