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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References:

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