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تسجيل مستخدم جديدInfinitely many small solutions to an elliptic PDE of variable exponent with a singular nonlinearity
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We prove the existence of infinitely many nonnegative solutions to the following nonlocal elliptic partial differential equation involving singularities begin{align} (-Delta)_{p(cdot)}^{s} u&=frac{lambda}{|u|^{gamma(x)-1}u}+f(x,u)~text{in}~Omega, onumber u&=0~text{in}~mathbb{R}^NsetminusOmega, onumber end{align} where $Omegasubsetmathbb{R}^N,, Ngeq2$ is a smooth, bounded domain, $lambda>0$, $sin (0,1)$, $gamma(x)in(0,1)$ for all $xinbar{Omega}$, $N>sp(x,y)$ for all $(x,y)inbar{Omega}timesbar{Omega}$ and $(-Delta)_{p(cdot)}^{s}$ is the fractional $p(cdot)$-Laplacian operator with variable exponent. The nonlinear function $f$ satisfies certain growth conditions. Moreover, we establish a uniform $L^{infty}(bar{Omega})$ estimate of the solution(s) by the Moser iteration technique.
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