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Existence and multiplicity of positive solutions for the \(p\)-Laplacian with nonlocal coefficient. (English) Zbl 1141.35029

Summary: We consider the Dirichlet problem with nonlocal coefficient given by \[ -a\Biggl(\int_\Omega |u|^q dx\Biggr)\,\Delta_p u= w(x) f(u) \] in a bounded, smooth domain \(\Omega\subset\mathbb{R}^n\) \((n\geq 2)\), where \(\Delta_p\) is the \(p\)-Laplacian, \(w\) is a weight function and the nonlinearity \(f(u)\) satisfies certain local bounds. In contrast with the hypotheses usually made, no asymptotic behavior is assumed on \(f\). We assume that the nonlocal coefficient \(a(\int_\Omega|u|^q dx)\) \((q\geq 1)\) is defined by a continuous and nondecreasing function \(a: [0,\infty)\to[0,\infty)\) satisfying \(a(t)> 0\) for \(t> 0\) and \(a(0)\geq 0\). A positive solution is obtained by applying the Schauder fixed point theorem. The case \(a(t)= t^{\gamma/q}\) \((0<\gamma< p-1)\) is considered as an example where asymptotic conditions on the nonlinearity provide the existence of a sequence of positive solutions for the problem with arbitrarily large sup norm.

MSC:

35J65 Nonlinear boundary value problems for linear elliptic equations
35B45 A priori estimates in context of PDEs
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