Multiplicity results for a degenerate elliptic equation with logistic reaction
This paper is concerned with multiplicity of positive solutions \(u\) of NEWLINE\[NEWLINE\lambda\text{ div}(|\nabla u|^{p-2}\nabla u)+ u^{q- 1}(1- u^r)= 0\quad\text{in }\OmegaNEWLINE\]NEWLINE with homogeneous Dirichlet boundary condition on \(\partial\Omega\). Here, \(\lambda\) is a positive parameter, \(2< p< q\), \(r>0\) and \(\Omega\) is a bounded domain in \(\mathbb{R}^N\) \((N\geq 2)\) with smooth boundary \(\partial\Omega\). The author proved in his previous paper [Proc. Am. Math. Soc. 129, 433--441 (2001; Zbl 0964.35066)], that there exists a positive number \(\Lambda\) such that if \(\lambda>\Lambda\), then this problem has no positive solution, and if \(\lambda\leq\Lambda\), then this problem has a maximal solution \(\overline u_\lambda\). Furthermore, it was proved that there exists a positive number \(\Lambda^*\in (0,\Lambda]\) such that for any \(\lambda\in (0,\Lambda^*)\), this problem has another positive solution \(u_\lambda\) with \(u_\lambda\leq\overline u_\lambda\) and \(u_\lambda\not\equiv\overline u_\lambda\). The main theorem of the present paper states that \(\Lambda^*\) equals \(\Lambda\).NEWLINENEWLINE By using the fact that \(0< u\leq 1\) in \(\Omega\) for all nonnegative nontrivial solution \(u\), the author considers the functional \(\Phi\) on \(W^{1,p}_0(\Omega)\) defined by NEWLINE\[NEWLINE\Phi(u)= {\lambda\over p} \int_\Omega|\nabla u|^p\,dx- \int_\Omega F(u)\,dx,NEWLINE\]NEWLINE where \(F(u)= \int^u_0 f(s)\,ds\) and \(f(s)= s^{q-1}(1- s^r)\) if \(s\in [0,\xi]\), \(=0\) if \(s\in (-\infty,0)\), and \(=\xi^{q-1}(1- \xi^r)\) if \(s\in (\xi,\infty)\), for \(\xi> 1\) fixed. To prove the theorem, he uses an extended mountain pass theorem due to \textit{P. Pucci} and \textit{J. Serrin} [J. Funct. Anal. 59, 185--210 (1984; Zbl 0564.58012)], which asserts that, if \(\Phi\) has a pair of local minima, then \(\Phi\) possesses a third critical point. For this purpose it is proved that the trivial solution \(u\equiv 0\) is a local minimizer of \(\Phi\) in \(W^{1,p}_0(\Omega)\) for every \(\lambda>0\), and that if the maximal solution \(\overline u_\lambda\) is isolated, then \(\overline u_\lambda\) is also a local minimizer of \(\Phi\) in \(W^{1,p}_0(\Omega)\) for every \(\lambda\in (0,\Lambda)\).
scientific article; zbMATH DE number 1657261
scientific article