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Regularity and multiplicity results for fractional $(p,q)$-Laplacian equations | Divya Goel
; Deepak Kumar
; K. Sreenadh
; | Date: |
1 Feb 2019 | Abstract: | This article deals with the study of the following nonlinear doubly nonlocal
equation:
egin{equation*}
(-Delta)^{s_1}_{p}u+a(-Delta)^{s_2}_{q}u = la a(x)|u|^{delta-2}u+
b(x)|u|^{r-2} u,; ext{ in }; Om, ; u=0 ext{ on } mathbb{R}^nsetminus
Om,
end{equation*}
where $Om$ is a bounded domain in $mathbb{R}^n$ with smooth boundary, $1<
de le qleq p<r leq p^{*}_{s_1}$, with $p^{*}_{s_1}=ds frac{np}{n-ps_1}$,
$0<s_2 < s_1<1$, $n> p s_1$ and $la, a>0$ are parameters. Here $ain
L^{frac{r}{r-de}}(Om)$ and $bin L^{infty}(Om)$ are sign changing
functions. We prove the $L^infty$ estimates, weak Harnack inequality and
Interior H"older regularity of the weak solutions of the above problem in the
subcritical case $(r<p_{s_1}^*).$ Also, by analyzing the fibering maps and
minimizing the energy functional over suitable
subsets of the Nehari manifold, we prove existence and multiplicity of weak
solutions to above
convex-concave problem. In case of $de=q$, we show the existence of
solution. | Source: | arXiv, 1902.0395 | Services: | Forum | Review | PDF | Favorites |
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