2021/06/14 by Gael Diebou Yomgne, Yomgne, Gael Diebou
Mathematics · Computer Science · #Nonlinear Partial Differential Equations #Advanced Mathematical Physics Problems #Advanced Mathematical Modeling in Engineering
paper · pdf · doi:10.48550/arxiv.2106.07567
We consider in this paper the nonlinear elliptic equation with Neumann\nboundary condition \
begincases
Delta u=a|u|m-1u
,
,
mbox\nin
,
,
rnp
dfrac
partial u
partial t=b|u|
eta-1u+f
,
,
mbox on\n
,
,
partial
rnp.
endcases For a,b\≠ 0,\nm>\(n+1)/(n-1), (n>1), \η=\(m+1)/(2) and small data f\∈\nL\(nq)/(n+1),\∞(\∂ rnp), q=\((n+1)(m-1))/(m+1) we prove\nthat the problem is solvable. More precisely, we establish existence,\nuniqueness and continuous dependence of solutions on the boundary data f in\nthe function space Xq\∞ where\n\
|u
|_
Xq
infty=
suptgt;0t^
fracn+1q-1
|u(
cdot,t)
|L
infty(
partial
rnp)+
|u
|_L^
fracq(m+1)2,
infty(
rnp)+
|
nablaν
|Lq,
infty(
rnp). As a direct consequence, we obtain the local\nregularity property C1,\νloc, \ν\∈ (0,1) of these solutions as\nwell as energy estimates for certain values of m. Boundary values decaying\nfaster than |x|-(m+1)/(m-1), x\∈ rn\∖ 0 yield solvability\nand this decay property is shown to be sharp for positive nonlinearities.\n Moreover, we are able to show that solutions inherit qualitative features of\nthe boundary data such as positivity, rotational symmetry with respect to the\n(n+1)-axis, radial monotonicity in the tangential variable and homogeneity.\nWhen a,b>0, the critical exponent mc for the existence of positive\nsolutions is identified, mc=(n+1)/(n-1).\n