Multiple Results for Critical Quasilinear Elliptic Systems Involving Concave-Convex Nonlinearities and Sign-Changing Weight Functions

Chang-Mu Chu ; Chun-Lei Tang

Bulletin of the Malaysian Mathematical Society, Tome 36 (2013) no. 3

Résumé

This paper is devoted to study the multiplicity of nontrivial nonnegative or positive solutions to the following systems \begin{eqnarray*}\label{iii} \left\{ \begin{array}{ll} -\triangle_p u=\lambda a_1(x)|u|^{q-2}u+b(x)F_u(u,v), \ \ \mbox{in}\ \ \Omega, \\ -\triangle_p v=\lambda a_2(x)|v|^{q-2}v+b(x)F_v(u,v), \ \ \mbox{in}\ \ \Omega,\\ u=v=0, \ \ \mbox{on}\ \ \partial\Omega, \end{array} \right. \end{eqnarray*} where $\Omega \subset R^{N}$ is a bounded domain with smooth boundary $\partial\Omega$; $1 q p N$, $p^{*}=\frac{Np}{N-p}$; $\triangle_{p} w=\mbox{div}(|\nabla w|^{p-2}\nabla w)$ denotes the $p$-Laplacian operator; $\lambda>0$ is a positive parameter; $a_i \in L^{\Theta}(\Omega)(i=1,\ 2)$ with $\Theta=\frac{p^{*}}{p^{*}-q}$ and $b\in L^{\infty}(\Omega)$ are allowed to change sign; $F\in C^{1}((R^+)^{2},R^+)$ is positively homogeneous of degree $p^{*}$, that is, $F(tz)=t^{p^{*}}F(z)$ holds for all $z\in (R^{+})^{2}$ and $t> 0 $, here, $R^{+}=[0,+\infty)$. The multiple results of weak solutions for the above critical quasilinear elliptic systems are obtained by using the Ekeland's variational principle and the mountain pass theorem.

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Classification : 35J50, 35J55, 35J92