2024/05/09 by Bal, Kaushik, Das, Stuti · 1 citation
#Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.2405.05832
We will prove multiplicity results for the mixed local-nonlocal elliptic equation of the form \beginsplit -Δpu+(-Δ)ps uamp;=\fracλuγ+ur \text in Ω,
uamp;gt;0 in Ω,
uamp;=0 \text in ℝn \backslash Ω; \endsplit where (-Δ)ps u(x)= cn,sP.V.∫ℝn\frac|u(x)-u(y)|p-2(u(x)-u(y))|x-y|n+sp d y, and -Δp is the usual p-Laplace operator. Under the assumptions that Ω is a bounded domain in ℝn with regular enough boundary, p>1, n> p, s∈(0,1), λ>0 and r∈(p-1,p^*-1) where p^* is the critical Sobolev exponent, we will show there exist at least two weak solutions to our problem for 0<γ<1 and some certain values of λ. Further, for every γ>0, assuming strict convexity of Ω, for p=2 and s∈(0,1/2), we will show the existence of at least two positive weak solutions to the problem, for small values of λ, extending the result of \citegaraingeometric. Here cn,s is a suitable normalization constant, and P.V. stands for Cauchy Principal Value.