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- W3166279683 abstract "We consider in this paper the nonlinear elliptic equation with Neumann boundary condition begin{align*} begin{cases} Delta u=a|u|^{m-1}u,,mbox{ in },,rnp dfrac{partial u}{partial t}=b|u|^{eta-1}u+f,,mbox{ on },,partialrnp. end{cases} end{align*} For $a,bneq 0$, $m>frac{n+1}{n-1}$, $(n>1)$, $eta=frac{m+1}{2}$ and small data $fin L^{q,infty}(partialrnp)$, $q=frac{n(m-1)}{m+1}$ we prove that the problem is solvable. More precisely, we establish existence, uniqueness and continuous dependence of solutions on the boundary data $f$ in the function space $X^{q}_{infty}$ where [|u|_{X^{q}_{infty}}=sup_{t>0}t^{frac{n}{q}}|u(cdot,t)|_{L^{infty}(partialrnp)}+|tr |_{L^{q,infty}(partialrnp)}+|u|_{L^{frac{(n+1)q}{n},infty}(rnp)}. ] As a direct consequence, we derive the local regularity property $C^{infty}_{loc}(rnp)$ of these solutions as well as energy estimates for certain values of $m$. Boundary values decaying faster than $|x|^{-(m+1)/(m-1)}$, $xin rnsetminus{0}$ yield solvability and this decay property is shown to be sharp for positive nonlinearities. Moreover, we are able to show that these solutions inherit qualitative features of the boundary data such as positivity (resp. negativeness), rotational symmetry with respect to the $(n+1)$-axis, radial monotonicity in the tangential variable and self-similarity. When $a,b>0$, the critical exponent $m_c$ for the existence of positive solutions is identified, $m_c=(n+1)/(n-1)$." @default.
- W3166279683 created "2021-06-22" @default.
- W3166279683 creator A5021428648 @default.
- W3166279683 date "2021-06-14" @default.
- W3166279683 modified "2023-09-27" @default.
- W3166279683 title "On a nonlinear Laplace equation related to the boundary Yamabe problem in the upper-half space" @default.
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