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Existence of positive solutions for some problems with nonlinear diffusion. (English) Zbl 0884.35039

Summary: We study the existence of positive solutions for problems of the type \[ -\Delta_pu(x) =u(x)^{q-1} h\bigl(x,u(x) \bigr),\;x\in\Omega, \quad u(x)=0,\;x\in \partial \Omega, \] where \(\Delta_p\) is the \(p\)-Laplace operator and \(p,q>1\). If \(p=2\), such problems arise in population dynamics. Making use of different methods (sub- and super-solutions and a variational approach), we treat the cases \(p=q\), \(p<q\) and \(p>q\), respectively. Also, some systems of equations are considered.

MSC:

35J65 Nonlinear boundary value problems for linear elliptic equations
35J20 Variational methods for second-order elliptic equations
35J55 Systems of elliptic equations, boundary value problems (MSC2000)
47J25 Iterative procedures involving nonlinear operators
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