Positive solutions for critical inhomogeneous elliptic problems in non-contractible domains. (English) Zbl 1155.35373
Summary: We investigate the existence of multiple solutions for critical inhomogeneous elliptic problem in non-contractible domains. We show that for \(f\in H^{-1}\) satisfying a suitable condition and the Dirichlet problem
\[ -\Delta u=|u|^{2^*-2}u+f, \quad x\in\Omega, \qquad u=0, \quad x\in\partial\Omega, \]
admits at least four positive solutions, where \(\Omega\supset B_{\frac 1\rho} \setminus \overline {B_\rho(0)}\), \(\overline\Omega\not\supset B_\rho(0)\) and \(\rho\) is sufficiently small.
\[ -\Delta u=|u|^{2^*-2}u+f, \quad x\in\Omega, \qquad u=0, \quad x\in\partial\Omega, \]
admits at least four positive solutions, where \(\Omega\supset B_{\frac 1\rho} \setminus \overline {B_\rho(0)}\), \(\overline\Omega\not\supset B_\rho(0)\) and \(\rho\) is sufficiently small.
MSC:
| 35J65 | Nonlinear boundary value problems for linear elliptic equations |
| 35J20 | Variational methods for second-order elliptic equations |
| 35J60 | Nonlinear elliptic equations |
Keywords:
positive solutions; critical Sobolev exponent; semilinear elliptic equation; Palais-Smale condition; Ljusternik-Schnirelman category; inhomogeneous problemReferences:
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