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Register a new userThe existence of a nontrivial weak solution to a double critical problem involving fractional Laplacian in ${R}^n$ with a Hardy term
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In this paper, we consider the existence of nontrivial weak solutions to a double critical problem involving fractional Laplacian with a Hardy term: begin{equation} label{eq0.1} (-Delta)^{s}u-{gamma} {frac{u}{|x|^{2s}}}= {frac{{|u|}^{ {2^{*}_{s}}(beta)-2}u}{|x|^{beta}}}+ big [ I_{mu}* F_{alpha}(cdot,u) big](x)f_{alpha}(x,u), u in {dot{H}}^s(R^{n}) end{equation} where $s in(0,1)$, $0leq alpha,beta<2s<n$, $mu in (0,n)$, $gamma<gamma_{H}$, $I_{mu}(x)=|x|^{-mu}$, $F_{alpha}(x,u)=frac{ {|u(x)|}^{ {2^{#}_{mu} }(alpha)} }{ {|x|}^{ {delta_{mu} (alpha)} } }$, $f_{alpha}(x,u)=frac{ {|u(x)|}^{{ 2^{#}_{mu} }(alpha)-2}u(x) }{ {|x|}^{ {delta_{mu} (alpha)} } }$, $2^{#}_{mu} (alpha)=(1-frac{mu}{2n})cdot 2^{*}_{s} (alpha)$, $delta_{mu} (alpha)=(1-frac{mu}{2n})alpha$, ${2^{*}_{s}}(alpha)=frac{2(n-alpha)}{n-2s}$ and $gamma_{H}=4^sfrac{Gamma^2(frac{n+2s}{4})} {Gamma^2(frac{n-2s}{4})}$. We show that problem (ref{eq0.1}) admits at least a weak solution under some conditions. To prove the main result, we develop some useful tools based on a weighted Morrey space. To be precise, we discover the embeddings begin{equation} label{eq0.2} {dot{H}}^s(R^{n}) hookrightarrow {L}^{2^*_{s}(alpha)}(R^{n},|y|^{-alpha}) hookrightarrow L^{p,frac{n-2s}{2}p+pr}(R^{n},|y|^{-pr}) end{equation} where $s in (0,1)$, $0<alpha<2s<n$, $pin[1,2^*_{s}(alpha))$, $r=frac{alpha}{ 2^*_{s}(alpha) }$; We also establish an improved Sobolev inequality. By using mountain pass lemma along with an improved Sobolev inequality, we obtain a nontrivial weak solution to problem (ref{eq0.1}) in a direct way. It is worth while to point out that the improved Sobolev inequality could be applied to simplify the proof of the main results in cite{NGSS} and cite{RFPP}.
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In this paper, we establish a new improved Sobolev inequality based on a weighted Morrey space. To be precise, there exists $C=C(n,m,s,alpha)>0$ such that for any $u,v in {dot{H}}^s(mathbb{R}^{n})$ and for any $theta in (bar{theta},1)$, it holds that begin{equation} label{eq0.3} Big( int_{ mathbb{R}^{n} } frac{ |(uv)(y)|^{frac{2^*_{s}(alpha)}{2} } } { |y|^{alpha} } dy Big)^{ frac{1}{ 2^*_{s} (alpha) }} leq C ||u||_{{dot{H}}^s(mathbb{R}^{n})}^{frac{theta}{2}} ||v||_{{dot{H}}^s(mathbb{R}^{n})}^{frac{theta}{2}} ||(uv)||^{frac{1-theta}{2}}_{ L^{1,n-2s+r}(mathbb{R}^{n},|y|^{-r}) }, end{equation} where $s !in! (0,1)$, $0!<!alpha!<!2s!<!n$, $2s!<!m!<!n$, $bar{theta}=max { frac{2}{2^*_{s}(alpha)}, 1-frac{alpha}{s}cdotfrac{1}{2^*_{s}(alpha)}, frac{2^*_{s}(alpha)-frac{alpha}{s}}{2^*_{s}(alpha)-frac{2alpha}{m}} }$, $r=frac{2alpha}{ 2^*_{s}(alpha) }$ and $y!=!(y,y) in mathbb{R}^{m} times mathbb{R}^{n-m}$. By using mountain pass lemma and (ref{eq0.3}), we obtain a nontrivial weak solution to a doubly critical system involving fractional Laplacian in $mathbb{R}^{n}$ with partial weight in a direct way. Furthermore, we extend inequality (ref{eq0.3}) to more general forms on purpose of studying some general systems with partial weight, involving p-Laplacian especially.
We establish the existence and multiplicity of positive solutions to the problems involving the fractional Laplacian: begin{equation*} left{begin{array}{lll} &(-Delta)^{s}u=lambda u^{p}+f(u),,,u>0 quad &mbox{in},,Omega, &u=0quad &mbox{in},,mathbb{R}^{N}setminusOmega, end{array}right. end{equation*} where $Omegasubset mathbb{R}^{N}$ $(Ngeq 2)$ is a bounded smooth domain, $sin (0,1)$, $p>0$, $lambdain mathbb{R}$ and $(-Delta)^{s}$ stands for the fractional Laplacian. When $f$ oscillates near the origin or at infinity, via the variational argument we prove that the problem has arbitrarily many positive solutions and the number of solutions to problem is strongly influenced by $u^{p}$ and $lambda$. Moreover, various properties of the solutions are also described in $L^{infty}$- and $X^{s}_{0}(Omega)$-norms.
In this paper we deal with the multiplicity of positive solutions to the fractional Laplacian equation begin{equation*} (-Delta)^{frac{alpha}{2}} u=lambda f(x)|u|^{q-2}u+|u|^{2^{*}_{alpha}-2}u, quadtext{in},,Omega, u=0,text{on},,partialOmega, end{equation*} where $Omegasubset mathbb{R}^{N}(Ngeq 2)$ is a bounded domain with smooth boundary, $0<alpha<2$, $(-Delta)^{frac{alpha}{2}}$ stands for the fractional Laplacian operator, $fin C(Omegatimesmathbb{R},mathbb{R})$ may be sign changing and $lambda$ is a positive parameter. We will prove that there exists $lambda_{*}>0$ such that the problem has at least two positive solutions for each $lambdain (0,,,lambda_{*})$. In addition, the concentration behavior of the solutions are investigated.
In this paper, we consider unsaturated poroelasticity, i.e., coupled hydro-mechanical processes in unsaturated porous media, modeled by a non-linear extension of Biots quasi-static consolidation model. The coupled, elliptic-parabolic system of partial differential equations is a simplified version of the general model for multi-phase flow in deformable porous media obtained under similar assumptions as usually considered for Richards equation. In this work, the existence of a weak solution is established using regularization techniques, the Galerkin method, and compactness arguments. The final result holds under non-degeneracy conditions and natural continuity properties for the non-linearities. The assumptions are demonstrated to be reasonable in view of geotechnical applications.
We are concerned with existence results for a critical problem of Brezis-Nirenberg Type involving an integro-differential operator. Our study includes the fractional Laplacian. Our approach still applies when adding small singular terms. It hinges on appropriate choices of parameters in the mountain-pass theorem
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