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Unbounded solutions of BVP for second order ODE with \(p\)-Laplacian on the half line. (English) Zbl 1274.34088

The paper studies the differential equation \[ (\rho (t)\varphi (x'(t)))'+f(t,x(t))=0, \quad t\in (0,\infty ) \eqno (1) \] subject to the boundary conditions \[ x(0)=\int _0^{\infty }g(s)x(s)\,\text{d}s +a,\quad \lim _{t\to \infty }\varphi ^{-1}(\rho (t))x'(t)=b, \eqno (2) \] where \(a,b\geq 0\), \(g: [0,\infty )\to [0,\infty )\) is continuous with \(\int _0^{\infty }g(s)\, \text{d}s <1\), \(f\:(0,\infty )\times [0,\infty )\to [0,\infty )\), \(\rho \: (0,\infty )\to (0,\infty )\) are continuous and may be singular at \(t=0\), and \(\varphi (x)=| x| ^{p-2}x\) with \(p>1\). The authors apply the Leggett-Williams fixed point theorem and derive a multiplicity result about the existence of at least three unbounded positive solutions of problem (1), (2). The proof is based on a special construction of a cone, which cannot be constructed by a standard way because possible solutions are not concave if \(\rho \not \equiv 1\). An illustrative example is shown as well.

MSC:

34B40 Boundary value problems on infinite intervals for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
34C11 Growth and boundedness of solutions to ordinary differential equations
34B16 Singular nonlinear boundary value problems for ordinary differential equations
47N20 Applications of operator theory to differential and integral equations
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