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On elliptic equations in \(\mathbb R^ N\) with critical exponents. (English) Zbl 0854.35037

Summary: We use variational arguments – namely Ekeland’s principle and the Mountain Pass Theorem – to study the equation \[ - \Delta u+ a(x) u= \lambda u^q+ u^{2^*- 1}\quad \text{in } \mathbb{R}^N. \] The main concern is overcoming compactness difficulties due both to the unboundedness of the domain \(\mathbb{R}^N\), and the presence of the critical exponent \(2^*= 2N/(N- 2)\).

MSC:

35J65 Nonlinear boundary value problems for linear elliptic equations
35J20 Variational methods for second-order elliptic equations
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