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Unbalanced $(p,2)$-fractional problems with critical growth | Deepak Kumar
; K. Sreenadh
; | Date: |
21 Jan 2020 | Abstract: | We study the existence, multiplicity and regularity results of non-negative
solutions of following doubly nonlocal problem: $$ (P_la) left{
egin{array}{lr}ds
quad (-Delta)^{s_1}u+a (-Delta)^{s_2}_{p}u = la a(x)|u|^{q-2}u+
left(int_{Om}frac{|u(y)|^r}{|x-y|^{mu}}~dy
ight)|u|^{r-2} u quad
ext{in}; Om,
quad quadquad quad u =0quad ext{in} quad mb R^nsetminus Om,
end{array}
ight. $$ where $Omsubsetmb R^n$ is a bounded domain with $C^2$
boundary $paOm$, $0<s_2 < s_1<1$, $n> 2 s_1$, $1< q<p< 2$, $1<r leq
2^{*}_{mu}$ with $2^{*}_{mu}=frac{2n-mu}{n-2s_1}$, $la,a>0$ and $ain
L^{frac{d}{d-q}}(Om)$, for some $q<d<2^{*}_{s_1}:=frac{2n}{n-2s_1}$, is a
sign changing function. We prove that each nonnegative weak solution of
$(P_la)$ is bounded. Furthermore, we obtain some existence and multiplicity
results using Nehari manifold method. | Source: | arXiv, 2001.7314 | Services: | Forum | Review | PDF | Favorites |
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