An inhomogeneous semilinear equation in entire space (Q1910867)

This paper deals with the inhomogeneous semilinear equation \[ \Delta u+ u^p+ f(x)= 0\quad \text{in}\quad \mathbb{R}^n,\tag{E} \] \(u> 0\), \(p> 1\), \(n\in \mathbb{N}\) with \(n\geq 3\), \(f\in C(\mathbb{R}^n)\) with \(f\geq 0\) everywhere in \(\mathbb{R}^n\), \(f\not\equiv 0\). Necessary conditions for the existence of a solution and a priori estimates are given as well as several existence results under various assumptions on \(f\). The author gives an existence theorem in the case \(p> n/(n- 2)\), where \(f\in C^{0, \gamma}(\mathbb{R}^n)\), \(0< \gamma\leq 1\), \(f\leq C_1(p)\cdot (1+ |x|^2)^{- {p\over p-1}}\) for some constant \(C_1(p)\), and a nonexistence result in the case \(p> n/(n- 2)\), \(f\geq C\cdot |x|^{- {2p\over p-1}}\) at \(\infty\) for \(C> C_2(p)\) (\(C_1(p)\) and \(C_2(p)\) are such that \(\lim_{p\to + \infty} C_2(p)/C_1(p)= 1\)). The proof of the nonexistence result is based on integral estimates, the use of spherical means and Jensen's inequality (the method also proves that there is no solution of (E) if \(p\leq n/(n- 2)\)). The proof of existence of solutions with slow decay (\(O(|x|^{- {2\over p- 1}})\) at \(\infty\)) relies on a super-subsolution method when \(p> n/(n- 2)\) and \(f\) not too large. A fast decay (\(O(|x|^{- (n+ 2)})\) at \(\infty\)) result is proved with the same method provided \(p\geq (n+ 2)/(n- 2)\), and with a contraction argument in the general case \((p> n/(n- 2))\), under various smallness assumptions on \(f\). An existence result for solutions in a neighborhood of the radial solution of \(- \Delta u_\alpha+ u^p_\alpha= 0\), \(u_\alpha(0)= \alpha> 0\) is also given in the case \(n\geq 12\), \(p\geq (n- 2)^2/((n- 2)^2- 8n)\) (there exists a radial solution \(u_\alpha\) only if \(p\geq (n+ 2)/(n- 2)\) and no solution to (E) such that \(u\geq u_\alpha\), \(\alpha> 0\) exists if \(n\leq 10\) or if \(n\geq 11\) and \(p< (n- 2)^2- 4n+ 8\sqrt{n- 1}\)).