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On the multiplicity of solutions of semilinear equations. (English) Zbl 0999.35028
From the introduction: We study exact multiplicity and uniqueness of solutions for problems on both balls and annular domains in \(\mathbb{R}^n\). On an annulus \(\Omega = \{x \mid A < |x|< B\}\) in \(\mathbb{R}^n\), \(n\geq 2\), we consider the problem \[ \Delta(u)+\lambda(u) = 0\text{ in }\Omega,\quad u = 0\text{ on }\partial\Omega.\tag{1.1} \] We study positive radially symmetric solutions of (1.1), depending on a positive parameter \(\lambda\). To get exact multiplicity results, we shall restrict our attention to the case of “thin” annulus. Set \(c_n=(2n-3)^{\frac{1}{n-2}}\) for \(n\geq 3\), and \(c_2= e^2\). We shall assume \[ B\leq c_n A.\tag{1.2} \] The crucial role in our study will be played by the linearized problem \[ \Delta w+\lambda f'(u)w = 0\text{ in }\Omega,\quad w=0\text{ on }\partial\Omega.\tag{1.3} \] In fact we proved in [P. Korman, Uniqueness and exact multiplicity results for two classes of semilinear problems, Nonlinear Anal. Theory Methods Appl. 31, 849–865 (1998; Zbl 0901.34027)] that in case \(f(u)\in C^2(\overline{\mathbb{R}_+})\) satisfies \[ f(u) > 0\text{ for\;all }u > 0\tag{1.4} \] then any nontrivial solution of (1.3) is of one sign. We used positivity of \(w(r)\) to prove several uniqueness and exact multiplicity results. In the present work we present a considerably easier proof of positivity of \(w(r)\) (see Corollary to Theorem 2.4). Moreover, we observe that positive solutions on annular domains are unimodular, and “tilted” to the left. This allowed us to considerably increase the width of the annulus on which uniqueness and exact multiplicity results hold.
We also consider a class of polynomial nonlinearities on balls in \(\mathbb{R}^2\). Using a transformation introduced by W.-M. Ni and R. D. Nussbaum [Uniqueness and nonuniqueness for positive ratial solutions of \(\Delta u+f(u,r)=0\), Commun. Pure Appl. Math. 38, 67–108 (1985; Zbl 0581.35021)], we are able to describe the global solution structure.

35J60 Nonlinear elliptic equations
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35J65 Nonlinear boundary value problems for linear elliptic equations
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