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Upper and lower solutions for the singular \(p\)-Laplacian with sign changing nonlinearities and nonlinear boundary data. (English) Zbl 1082.34022

The paper stydies the boundary value problem \[ -\bigl(\varphi_p(u')\bigr)'=q(t)f(t,u),\quad 0<t<1,\quad u(0)=0,\quad \Psi\bigl(u(1)\bigr)+u'(1)=0, \] which may be singular at \(u=0,\;t=0\) and \(t=1.\) Here \(\varphi_p(s)=| s| ^{p-2}s,\;p>1\), \(q\in C(0,1)\cap L^1[0,1]\) with \(q>0\) on \((0,1)\), \(f\in C([0,1]\times(0,\infty),\mathbb{R})\) is allowed to change sign and \(\Psi\in C(\mathbb{R},\mathbb{R})\) may be nonlinear. A general existence theorem is obtained under the existence of lower and upper solutions. To prove it, the singular problem by a sequence of nonsingular problems. The solvability of each approximating problem is approximated and the considered singular problem follows from Schauder’s fixed-point theorem and the Arzela-Ascoli theorem, respectively. Constructing lower and upper solutions and using the obtained general existence theorem, the authors establish a solution \(u\in C[0,1]\cap C^1(0,1]\) such that \(u(t)>0\) for \(t\in(0,1]\) and \(\varphi_p(u')\in C^1(0,1).\)

MSC:

34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
34B16 Singular nonlinear boundary value problems for ordinary differential equations
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References:

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