Existence, multiplicity and regularity of solutions of elliptic problem involving non-local operator with variable exponents and concave-convex nonlinearity

Reshmi Biswas, Sweta Tiwari

Published 2018-10-30Version 1

In this paper, first we introduce the variable exponent fractional Sobolev space $W^{s(x,y),q(x),p(x,y)}(\Omega).$ Then, using variational methods we study the existence and multiplicity of solution of the following variable order non-local problem involving concave-convex type nonlinearity: \begin{align*} (-\Delta)_{p(\cdot)}^{s(\cdot)} u(x)&=\lambda\mid u(x)\mid^{\alpha(x)-2}u(x)dx+f(x,u),\hspace{3mm} x\in \Omega, % u&>0 ,\hspace{15mm} x\in \Omega, u&=0 ,\hspace{38mm}x\in \mathcal{C}\Omega:=\mathbb R^n\setminus\Omega, \end{align*} where $\lambda>0,$ $p\in C(\overline{\Omega}\times \overline{\Omega},(1,\infty))$, $s\in C(\overline{\Omega}\times\overline{\Omega}, (0,1))$ and $q,\alpha\in C(\overline{\Omega},(1,\infty))$ and $f:\Omega\times\mathbb R\rightarrow[0,\infty)$ is a Carat\'{e}odory function with subcritical growth. We also prove the uniform estimate for the solution of the above problem.