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A multiplicity results for a singular equation involving the \(p(x)\)-Laplace operator. (English) Zbl 1365.35032

Summary: The purpose of this paper is to study the singular problem involving the \(p(x)\)-Laplace operator: \[ (\mathrm{P}_\lambda)=\begin{cases} -\Delta_{p(x)}u=\frac{\lambda}{u^{\delta(x)}}+f(x,u)\quad\text{in }\Omega,\\ u>0\quad\text{in }\Omega,\\ u=0,\quad\text{on }\partial\Omega.\end{cases} \] where \(\Omega\subset\mathbb{R}^N,(N\geq2)\) be a bounded domain with \(C^2\) boundary, \(\lambda\) is a positive parameter \(p(x)\), \(\delta(x)\) and \(f(x,u)\) are assumed to satisfy the assumptions \((\mathrm{H}0)-(\mathrm{H}4)\) in the Introduction. We employ variational techniques in order to show the existence of a number \(\Lambda\in(0,\infty)\) such that problem \((P_\lambda)\) has two solutions for \(\lambda\in(0,\Lambda)\), one solution for \(\lambda=\Lambda\) and no solutions for \(\lambda>\Lambda\). To obtain multiple (at least two distinct, positive) solutions of problem \((P_\lambda)\), we need to prove two new results: a regularity result for solutions to problem \((P_\lambda)\) in \(C^{1,\alpha}(\overline{\Omega})\) with some \(\alpha\in(0,1)\), and a strong comparison principle.

MSC:

35J60 Nonlinear elliptic equations
35A15 Variational methods applied to PDEs
35R11 Fractional partial differential equations
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