scholarly journals Multiplicity of Positive Solutions for Some Quasilinear Elliptic Equation in RNwith Critical Sobolev Exponent

1997 ◽  
Vol 140 (1) ◽  
pp. 106-132 ◽  
Author(s):  
Pavel Drábek ◽  
Yin Xi Huang
Keyword(s):  
Elliptic Equation ◽  
Positive Solutions ◽  
Quasilinear Elliptic Equation ◽  
Critical Sobolev Exponent ◽  
Sobolev Exponent ◽  
Multiplicity Of Positive Solutions ◽  
Quasilinear Elliptic

scholarly journals Multiplicity of Positive Solutions for ap-q-Laplacian Type Equation with Critical Nonlinearities

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Tsing-San Hsu ◽  
Huei-Li Lin
Keyword(s):  
Positive Solutions ◽  
Type Equation ◽  
Critical Sobolev Exponent ◽  
Critical Nonlinearity ◽  
Bounded Smooth Domain ◽  
Laplacian Equation ◽  
Critical Nonlinearities ◽  
Bounded Smooth ◽  
Sobolev Exponent ◽  
Multiplicity Of Positive Solutions

We study the effect of the coefficientf(x)of the critical nonlinearity on the number of positive solutions for ap-q-Laplacian equation. Under suitable assumptions forf(x)andg(x), we should prove that for sufficiently smallλ>0, there exist at leastkpositive solutions of the followingp-q-Laplacian equation,-Δpu-Δqu=fxu|p*-2u+λgxu|r-2u  in  Ω,u=0  on  ∂Ω,whereΩ⊂RNis a bounded smooth domain,N>p,1<q<N(p-1)/(N-1)<p≤max⁡{p,p^*-q/(p-1)}<r<p^*,p^*=Np/(N-p)is the critical Sobolev exponent, andΔsu=div(|∇u|s-2∇uis thes-Laplacian ofu.

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