Multiplicity of positive solutions for nonlinear elliptic equations on annulus. (English) Zbl 0757.35006

Summary: We study a semilinear elliptic equation \[ \Delta u+\lambda f(u)=0\quad\text{in }\Omega, \qquad\text{and}\qquad u=0 \quad\text{on }\partial\Omega, \] where \(\Omega\) is an annulus in \(\mathbb{R}^ n\), \(n\geq 2\), and \(f\) is positive, strictly increasing and strictly convex on \(\mathbb{R}^ +\). We prove that if \(f\) satisfies \[ (1+\varepsilon)f(u)\leq uf'(u)\leq(n/(n-2))f(u)\qquad\text{for } u>0,\quad \varepsilon>0\quad\text{and }n\geq 4, \] then there exists \(\lambda^*>0\) such that the equations have exactly two positive radial solutions for each \(\lambda\in(0,\lambda^*)\), exactly one for \(\lambda=\lambda^*\) and none for \(\lambda>\lambda^*\). Existence of many positive nonradial solutions are also obtained by using Nehari type variational method provided that the annulus is narrow enough.


35B32 Bifurcations in context of PDEs
35J65 Nonlinear boundary value problems for linear elliptic equations
35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
35J60 Nonlinear elliptic equations
35J20 Variational methods for second-order elliptic equations