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Positive solutions for infinite semipositone problems with falling zeros. (English) Zbl 1190.35095

Summary: We consider the positive solutions to the singular problem
\[ \begin{cases} -\Delta u=au-f(u)- \frac{c}{u^\alpha} &\text{in }\Omega,\\ u=0 &\text{on }\partial\Omega, \end{cases}\tag{P} \]
where \(0<\alpha<1\), \(a>0\) and \(c>0\) are constants, \(\Omega\) is a bounded domain with smooth boundary and \(f:[0,\infty)\to\mathbb R\) is a continuous function. We assume that there exist \(M>0\), \(A>0\), \(p>1\) such that \(au-M\leq f(u)\leq Au^p\), for all \(u\in[0,\infty)\). A simple example of \(f\) satisfying these assumptions is \(f(u)=u^p\) for any \(p>1\). We use the method of sub-supersolutions to prove the existence of a positive solution of (P) when \(a> \frac{2\lambda_1}{1+\alpha}\) and \(c\) is small. Here \(\lambda_1\) is the first eigenvalue of operator \(-\Delta\) with Dirichlet boundary conditions. We also extend our result to classes of infinite semipositone systems.

MSC:

35J61 Semilinear elliptic equations
35J25 Boundary value problems for second-order elliptic equations
35J20 Variational methods for second-order elliptic equations
35B09 Positive solutions to PDEs
35J91 Semilinear elliptic equations with Laplacian, bi-Laplacian or poly-Laplacian
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