scholarly journals Ground states of the quasilinear problems involving the critical Sobolev exponent and potentials

2009 ◽  
pp. 349-366
Author(s):  
Dongsheng Kang
Keyword(s):  
Ground States ◽  
Critical Sobolev Exponent ◽  
Quasilinear Problems ◽  
Sobolev Exponent

Construction of dipole type singular solutions for a biharmonic equation with critical Sobolev exponent

2002 ◽  
Vol 2 (4) ◽  
Author(s):  
Sarni Baraket
Keyword(s):  
Fourth Order ◽  
Weak Solutions ◽  
Biharmonic Equation ◽  
Critical Sobolev Exponent ◽  
Singular Solutions ◽  
Invariant Equation ◽  
Conformally Invariant ◽  
Sobolev Exponent

AbstractIn this paper, we construct positive weak solutions of a fourth order conformally invariant equation on S

scholarly journals On Critical p-Laplacian Systems

2017 ◽  
Vol 17 (4) ◽  
pp. 641-659
Author(s):  
Zhenyu Guo ◽  
Kanishka Perera ◽  
Wenming Zou
Keyword(s):  
Critical Sobolev Exponent ◽  
Laplacian Operator ◽  
Existence And Multiplicity ◽  
Nonnegative Solutions ◽  
Least Energy Solution ◽  
Existence And Nonexistence ◽  
Alpha Beta ◽  
Beta 2 ◽  
Sobolev Exponent ◽  
Least Energy

AbstractWe consider the critical p-Laplacian system\left\{\begin{aligned} &\displaystyle{-}\Delta_{p}u-\frac{\lambda a}{p}\lvert u% \rvert^{a-2}u\lvert v\rvert^{b}=\mu_{1}\lvert u\rvert^{p^{\ast}-2}u+\frac{% \alpha\gamma}{p^{\ast}}\lvert u\rvert^{\alpha-2}u\lvert v\rvert^{\beta},&&% \displaystyle x\in\Omega,\\ &\displaystyle{-}\Delta_{p}v-\frac{\lambda b}{p}\lvert u\rvert^{a}\lvert v% \rvert^{b-2}v=\mu_{2}\lvert v\rvert^{p^{\ast}-2}v+\frac{\beta\gamma}{p^{\ast}}% \lvert u\rvert^{\alpha}\lvert v\rvert^{\beta-2}v,&&\displaystyle x\in\Omega,\\ &\displaystyle u,v\text{ in }D_{0}^{1,p}(\Omega),\end{aligned}\right.where {\Delta_{p}u:=\operatorname{div}(\lvert\nabla u\rvert^{p-2}\nabla u)} is the p-Laplacian operator defined onD^{1,p}(\mathbb{R}^{N}):=\bigl{\{}u\in L^{p^{\ast}}(\mathbb{R}^{N}):\lvert% \nabla u\rvert\in L^{p}(\mathbb{R}^{N})\bigr{\}},endowed with the norm {{\lVert u\rVert_{D^{1,p}}:=(\int_{\mathbb{R}^{N}}\lvert\nabla u\rvert^{p}\,dx% )^{\frac{1}{p}}}}, {N\geq 3}, {1<p<N}, {\lambda,\mu_{1},\mu_{2}\geq 0}, {\gamma\neq 0}, {a,b,\alpha,\beta>1} satisfy {a+b=p}, {\alpha+\beta=p^{\ast}:=\frac{Np}{N-p}}, the critical Sobolev exponent, Ω is {\mathbb{R}^{N}} or a bounded domain in {\mathbb{R}^{N}} and {D_{0}^{1,p}(\Omega)} is the closure of {C_{0}^{\infty}(\Omega)} in {D^{1,p}(\mathbb{R}^{N})}. Under suitable assumptions, we establish the existence and nonexistence of a positive least energy solution of this system. We also consider the existence and multiplicity of the nontrivial nonnegative solutions.

Sign in / Sign up

Export Citation Format

Share Document