Global existence of solutions of mixed sublinear-superlinear differential equations
Let \(L_ n\) be the general disconjugate operator \[ (1)\quad L_ n=\frac{1}{p_ n(t)} \frac{d}{dt} \frac{1}{p_{n-1}(t)} \frac{d}{dt}... \frac{d}{dt} \frac{1}{p_ 1(t)} \frac{d}{dt} \frac{\cdot}{p_ 0(t)} (n\geq 2), \] where \(p_ i: [0,\infty)\to (0,\infty)\) \((0\leq i\leq n)\) are continuous and \(\int^{\infty}_{0}p_ i(t)dt=\infty\) \((1\leq i\leq n- 1)\). Consider the mixed sublinear-superlinear differential equation \[ (2)\quad L_ ny+[a(t)| y|^{\alpha}+b(t)| y|^{\beta}]sgn y=0\quad (t>0), \] where \(0<\alpha <1\), \(\beta >1\), and \(a,b:[0,\infty)\to {\mathbb{R}}\) are continuous. The main result is a condition which implies that (2) has a solution \(\hat y\) which is defined on the entire interval \([0,\infty)\) and behaves as \(t\to \infty\) like a prescribed solution of the unperturbed equation (3) \(L_ ny=0\) \((t>0)\). To be specific, let \(I_ 0=1\) and \(I_ j(t:q_ j,...,q_ 1)= \int^{t}_{0}q_ j(\tau)I_{j-1}(\tau;q_{j-1},...,q_ 1)d\tau;\) then the functions \(y_ i(t)=p_ 0(t)I_{i-1}(t;p_ 1,...,p_{i-1})\) and \(z_ i(t)=p_ n(t)I_{n-i}(t;p_{n-1},...,p_ i)\) \((1\leq i\leq n)\) form fundamental systems, respectively, for (3) and its formal adjoint \(L^*_ nz=0\), where \(L^*_ n\) is obtained by reversing the order of \(p_ 0,...,p_ n\) in (1). Theorem. Let \(A=\int^{\infty}_{0}| a(t)| y_ i^{\alpha}(t)z_ i(t)dt\) and \(B=\int^{\infty}_{0}| b(t)y_ i^{\beta}(t)z_ i(t)dt\) be finite for some i (1\(\leq i\leq n)\) and \(A^{\mu}B^{\nu}(\rho^{\mu}+\rho^{-\nu})<1\) where \(\mu = (\beta- 1)/(\beta-\alpha),\nu = (1-\alpha)/(\beta -\alpha),\) and \(\rho =(1- \alpha)/(\beta-1).\) Then, if c(\(\neq 0)\) is a suitably chosen constant, (2) has a solution \(\hat y\) which is defined on the entire interval \([0,\infty)\) such that \(\lim_{t\to \infty}\hat y(t)/y_ i(t)=c.\) Taking \(p_ 0=...=p_ n=1\) here yields a new global existence theorem for the mixed sublinear-superlinear equation \(y^{(n)} + [a(t)| y|^{\alpha} + b(t)| y|^{\beta}]sgn y=0.\) The theorem is also applied to obtain conditions for the mixed sublinear-superlinear elliptic equation \(\Delta u+\phi (| x|)u^{\lambda} + \psi(| x|)u^{\mu}=0\) \((x\in {\mathbb{R}}^ N\), \(N\geq 3)\) to have a radially symmetric solution \(\hat u\) which is defined and positive for all \(x\) in \({\mathbb{R}}^ N\), and has the asymptotic behavior \(\lim_{| x| \to \infty}| x|^{N-2}\hat u(x)=c(>0)\).
mixed sublinear-superlinear differential equation
scientific article; zbMATH DE number 4008857