Positive solutions for a class of nonlinear elliptic problems
The author considers the following problem: \[ \Delta u+f(u)=0\;\text{in } \Omega, \quad u=0\;\text{on }\Gamma_1,\quad u=b \text{ on } \Gamma_2, \tag{1} \] where \(f:\mathbb{R}^+\to \mathbb{R}^+\), \(\Omega\) is a bounded domain in \(\mathbb{R}^N\) with a hole, \(\Gamma_1(\Gamma_2)\) is the outer (inner) boundary, \(b\) is a positive number. Theorem 1. Let \(f\) be continuous and \(\lim_{x\to 0} \frac{f(x)}{x}=0, \lim_{x\to \infty} \frac{f(x)}{x}=\infty.\) Then there exists a positive number \(b^*\) such that problem (1) has a positive solution for \(b<b^*\) and no solution for \(b>b^*\). Theorem 2. Let \(f\) be locally Lipschitz continuous, convex and \(\lim_{x\to 0} \frac{F(x)}{x^2}=0\), \(F(t):=\int_0^t f(s) ds\); \(\lim_{x\to \infty} \frac{f(x)}{x}=\infty.\) Then there exists a positive number \(b^*\) such that the problem (1) with \(\Gamma_1=\{x\mid |x|=r>0\}\), \(\Gamma_2=\{x\mid |x|=R>r\}\) has at least two positive radially symmetric solutions for \(b<b^*\), at least one for \(b=b^*\) and none for \(b>b^*\). Theorem 1 extends result by \textit{M. G. Lee} and \textit{S. S. Lin} [J. Math. Anal. Appl. 181, 348-361 (1994; Zbl 0791.35045)] onto the case of nonconvex and nonsmooth functions \(f\). Theorem 2 extends and complements results by \textit{C. Bandle} and \textit{L. A. Peletier} [Math. Ann. 280, 1-19 (1988; Zbl 0619.35044)] formulated for \(f(u)=u^{(N+2)/(N-2)}\).
scientific article; zbMATH DE number 1310256