Existence of Positive Solution for Kirchhoff Problems
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Abstract
In this work, we study the following Kirchhoff type problem where \(p\ge 2\), \(\Omega \) is a regular bounded domain in \(\mathbb {R}^N\), \((N\ge 3)\). Firstly, for \(p>2\), we prove under some appropriate conditions on the singularity and the nonlinearity the existence of nontrivial weak solution to this problem. For \(p=2\), we show, under supplementary condition, the positivity of this solution. Moreover, in the case \(\lambda =0\) we prove an uniqueness result. We use the variational method to prove our main results.
$$\begin{aligned} \begin{gathered} -\Big (a+b\int _{\Omega }|\nabla u|^pdx\Big )\Delta _p u =g(x)u^{-\gamma }+\lambda f(x,u),\quad \text {in }\Omega , \\ u=0, \quad \text {on }\partial \Omega , \end{gathered} \end{aligned}$$
Keywords
Kirchhoff type equation Singularity problem Variational methods Resonance Positive solution Mountain pass lemmaMathematics Subject Classification
35B09 35B33 35J20References
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