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Multiple positive solutions of strongly indefinite systems with critical Sobolev exponents and data that change sign. (English) Zbl 1163.35376
Summary: We study the existence of multiple positive solutions to some Hamiltonian elliptic systems: (*) $-\Delta v=\lambda u+|u|^{p-1}u+ \varepsilon f(x)$, $-\Delta u=\mu v+|v|^{q-1}v+ \varepsilon g(x)$ in $\Omega$; $u>0$, $v>0$ in $\Omega$; $u=v=0$ on $\partial\Omega$, where $\Omega$ is a smooth bounded domain in $\Bbb R^N$ $(N\ge3)$; $f,g\in C^1(\overline{\Omega})$; $p,q>1$; $\lambda,\mu\in\Bbb R$. For the subcritical and critical cases, we prove that problem (*) has at least two positive solutions for any $\varepsilon\in(0,\varepsilon^*)$ and has no positive solutions for any $\varepsilon> \varepsilon^*$ $(\ge\varepsilon^*)$. In the supercritical case, we find that the existence of solutions of problem (*) for $\lambda=\mu=0$ is closely related to the existence of nonnegative solutions of some linear elliptic system.

35J60Nonlinear elliptic equations
35B33Critical exponents (PDE)
Full Text: DOI
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