Existence and multiplicity of positive solutions for semilinear elliptic systems with Sobolev critical exponents. (English) Zbl 1176.35064

Summary: We study the existence and multiplicity of positive solutions to the system \(-\Delta u= \frac{\partial F}{\partial u}(u,v)+\varepsilon g(x)\), \(-\Delta v= \frac{\partial F}{\partial v}(u,v)+\varepsilon h(x)\) in \(\Omega\); \(u,v>0\) and \(u=v=0\) on \(\partial\Omega\), where \(\Omega\) is a bounded smooth domain in \(\mathbb R^N\); \(F\in C^1((\mathbb R^+)^2,\mathbb R^+)\) is positively homogeneous of degree \(\mu\); \(g,h\in C^1(\overline{\Omega})\setminus\{0\}\); and \(\varepsilon\) is a positive parameter. Using sub-supersolution method, we prove the existence of positive solutions for the above problem. By means of the variational approach, we prove the multiplicity of positive solutions for the above problem with \(\mu\in(2,2^*]\).


35J57 Boundary value problems for second-order elliptic systems
35J61 Semilinear elliptic equations
35J50 Variational methods for elliptic systems
35B09 Positive solutions to PDEs
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