Classical and weak solutions of a singular semilinear elliptic problem. (English) Zbl 0880.35043

Authors’ summary: The singular semilinear elliptic equation \(\Delta u+p(x)f(u)= 0\) is shown to have a unique positive classical solution in \(\mathbb{R}^n\) that decays to zero at \(\infty\) if \(p(x)\) is simply a nontrivial nonnegative continuous function satisfying \(\int^\infty_0 t(\max_{|x|=t}p(x))dt<\infty\), provided \(f\) is a nonincreasing continuously differentiable function on \((0,\infty)\). It is also shown that the equation has a unique weak \(H^1_0\)-solution on a bounded domain provided \(\int^\varepsilon_0 f(s)ds<\infty\) and \(p(x)\in L^2\).


35J60 Nonlinear elliptic equations
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