Exact multiplicity of positive solutions for a class of semilinear problem. II
The authors study positive solutions to the semilinear problem: \[ \begin{cases} \Delta u + \lambda f(u) = 0,& \quad \text{ in }B^n, \\ u = 0,& \quad \text{ on }B^n, \end{cases} \tag{1} \] on the unit ball \(B^n\) in \({\mathbb R}^n\) with \(n\geq 1\), where \(\lambda\) is a positive parameter and \(f\) is a smooth function. This problem has a very long history and many contributions have been made in the last several decades on the existence and multiplicity of positive solutions. But a complete description on the global structure of the positive solution set of (1) is very hard to obtain even for certain typical classes of nonlinearities. The authors have obtained interesting results on this problem. Using the result of Gidas-Ni-Nirenberg on radial symmetry of positive solutions of (1), this problem is reduced to positive solutions of the following ordinary differential equation: \[ \begin{cases} u'' + {{n-1}\over r}u' + \lambda f(u) = 0, &\quad r\in (0,1), \\ u'(0) = u(1) = 0. &\quad \end{cases} \tag{2} \] Then using bifurcation analysis methods together with a Morse index estimate, the authors developed a unified approach to obtain the exact multiplicity of the positive solutions for several classes of nonlinear functions \(f(u)\), and the precise shape of the global bifurcation diagrams. The authors discovered that the shape of the bifurcation curve depends on the shape of the graph of the function \(f(u)/u\) and the growth rate of \(f\). Typical examples of the nonlinear function \(f(u)\) treated in this paper includes (A) \(f(u) = u^p\) with \(0<p<1\) or \(1<p<(n+2)/(n-2)\), (B) \(f(u) = \sqrt{u^2+2u}\), (C) \(f(u) = -u+u^p-c\) with \(c\geq 0\) and \(1<p<(n+2)/(n-2)\), (D) \(f(u) = u(u-b)(c-u)\) with \(0<2b<c\), (E) \(f(u) = u^p - u^q\) with \(0<p<q\), (F) \(f(u) = u^p + u^q\) with \(0<q<1<p\leq n/(n-2)\). The authors also give a partial answer to a conjecture of \textit{P. L. Lions} proposed [SIAM Rev. 24, 441-467 (1982; Zbl 0511.35033)], which claims that the structure of the set \(\{(\lambda,u)\}\) of positive solutions of (1) is similar to the structure of the solution set \(\{(\lambda,u)\}\) of an algebraic equation \[ \lambda_1 u = \lambda f(u), \] where \(\lambda_1\) is the first eigenvalue of \(-\Delta\) on \(H_1^0(B^n)\). For part I see J. Differ. Equations 146, No. 1, 121-156 (1998; Zbl 0918.35049)].
scientific article; zbMATH DE number 1384461
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