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Liang, Sihua,Zhang, Jihui,Fan, Fan The Korean Society for Computational and Applied M 2010 Journal of applied mathematics & informatics Vol.28 No.5
In this paper, we study the following p&q-Laplacian problem with critical Sobolev and Hardy exponents {$-{\Delta}_pu-{\Delta}_qu={\mu}\frac{{\mid}u{\mid}^{p^*(s)-2}u}{{\mid}x{\mid}^s}+{\lambda}f(x,\;u)$, in $\Omega$, u=0, on $\Omega$, where ${\Omega}\;{\subset}\;\mathbb{R}^{\mathbb{N}}$ is a bounded domain and ${\Delta}_ru=div({\mid}{\nabla}u{\mid}^{r-2}{\nabla}u)$ is the r-Laplacian of u. By using the variational method and concentration-compactness principle, we obtain the existence of infinitely many small solutions for above problem which are the complement of previously known results.
INFINITELY MANY SMALL SOLUTIONS FOR THE ρ(χ)-LAPLACIAN OPERATOR WITH CRITICAL GROWTH
Chenxing Zhou,Sihua Liang 한국전산응용수학회 2014 Journal of applied mathematics & informatics Vol.32 No.1
In this paper, we prove, in the spirit of [3, 12, 20, 22, 23], the existence of infinitely many small solutions to the following quasilinear elliptic equation −Δp(χ)u+|u|p(χ)−2u = |u|q(χ)−2u+f(χ,u) in a smooth bounded domain Ω of RN. We also assume that {q(x) = p*(x)} ≠∅, where p*(χ) = Np(χ)=(N −p(χ)) is the critical Sobolev exponent for variable exponents. The proof is based on a new version of the symmetric mountainpass lemma due to Kajikiya [22], and property of these solutions are also obtained.
INFINITELY MANY SMALL SOLUTIONS FOR THE p(x)-LAPLACIAN OPERATOR WITH CRITICAL GROWTH
Zhou, Chenxing,Liang, Sihua The Korean Society for Computational and Applied M 2014 Journal of applied mathematics & informatics Vol.32 No.1
In this paper, we prove, in the spirit of [3, 12, 20, 22, 23], the existence of infinitely many small solutions to the following quasilinear elliptic equation $-{\Delta}_{p(x)}u+{\mid}u{\mid}^{p(x)-2}u={\mid}u{\mid}^{q(x)-2}u+{\lambda}f(x,u)$ in a smooth bounded domain ${\Omega}$ of ${\mathbb{R}}^N$. We also assume that $\{q(x)=p^*(x)\}{\neq}{\emptyset}$, where $p^*(x)$ = Np(x)/(N - p(x)) is the critical Sobolev exponent for variable exponents. The proof is based on a new version of the symmetric mountainpass lemma due to Kajikiya [22], and property of these solutions are also obtained.