Positive solutions for Kirchhoff-type equations with critical exponent in \(\mathbb{R}^N\). (English) Zbl 1319.35021
Summary: In this paper, we study the following nonlinear Kirchhoff-type equation
\[
\begin{cases} -\left(a + b \int\limits_{\mathbb R^N}| \nabla u|^2 d x\right) \triangle u = | u |^{2^\ast - 2} u + \mu h(x),\quad x \in \mathbb R^N \\ u \in D^{1, 2}(\mathbb R^N), \end{cases}\eqno{(0.1)}
\]
where \(a \geq 0\), \(b > 0\), \(N \geq 3\), \(2^\ast = \frac{2 N}{N - 2}\), \(\mu \geq 0\) and \(h \in L^{\frac{2^\ast}{2^\ast - 1}}(\mathbb R^N) \backslash \{0 \}\) is nonnegative. By using the variational method, we obtain the existence of positive solutions for Equation (0.1), under some assumptions on \(a\), \(b\), \(\mu\).
MSC:
| 35J20 | Variational methods for second-order elliptic equations |
| 35B09 | Positive solutions to PDEs |
| 35B33 | Critical exponents in context of PDEs |
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