Existence and uniqueness for a class of quasilinear elliptic boundary value problems. (Q1412360)

The authors prove the existence and uniqueness of positive solutions for the quasilinear boundary value problem \[ (r^{N-1}\phi(u'))'=-\lambda r^{N-1}f(u),\quad u'(0)=u(1)=0, \leqno(1) \] where \(\phi(x)=| x| ^{p-2}x\) and \(\lambda\) is a real parameter. Under the assumptions \( f:[0,\infty)\to \mathbb{R}\) is continuous, of class \(C^1\) on \((0,\infty)\), nondecreasing, with \[ \lim_{x\to \infty} f(x)>0,\quad \lim_{x\to \infty}{f(x)\over x^{p-1}}=0, \] the existence of a positive solution is proved. If \(f(0)>0\), then a positive solution exists for all \(\lambda >0\) and if \(f(0)\leq 0\), then a positive solution exists for \(\lambda\) large. If moreover \[ \liminf_{x\to 0^+}{f(x)\over x^{p-1}}\not= 0, \quad \limsup_{x\to \infty}{xf'(x)\over f(x)}<p-1, \quad \limsup_{x\to 0^+}xf'(x)<\infty, \] then the uniqueness result for (1) is valid. Note, that \(f(x)\over x^{p-1}\) may not be decreasing on \((0,\infty)\).