Existence and nonexistence of sign-changing solutions for singular semilinear equations on exterior domains
In this paper, the author proves existence and nonexistence of radial solutions \(u\in C^1[R,\infty)\cap C^0[R,\infty)\) of the problem \[ \Delta u+K(r)f(u)=0\text{ in }\mathbb{R}^{N}\backslash B_{r},\, u=0\text{ on }\partial B_{R},\, u\to 0\text{ as }|x|\to\infty \] on \([R,\infty)\) using some of the following assumptions (i) \(f(u)=-(1/(|u|^{q-1}u))+g(u)\), \(f:\mathbb{R}\backslash 0\to\mathbb{R}\) is locally Lipschits, \(0<q<1\), \(g(0)=0\), \(f\) is odd, \(f<0\) on \((0,\beta)\), \(f>0\) on \((\beta,\infty)\), (ii) \(f(u)=|u|^{p-1}u+g_1(u)\), where \(\lim_{u\to\infty}((|g_1(u)|)/(|u|^{p}))=0\), \(p>1\), (iii) \(F(u)=\int_0^{u}f(s)ds(<0\) bounded below on \((0,\gamma),)/(>0\) on \((\gamma,\infty),)\) (iv) \(0<K(r)<C_1/r^\alpha\), \(\alpha>N+q(N-2)\), \(K(r)\) is continuously differentiable on \([R,\infty)\), \((v) 2(N-1)+((rK')/K)\geq 0\), where \(N>2\), \(B_{R}\) is the ball of radius \(R>0\) centered at the origin in \(\mathbb{R}^{N}\).
scientific article; zbMATH DE number 7472465