Resonance problems for Kirchhoff type equations
The authors consider the following Kirchhoff type problem with Dirichlet boundary condition \[ \begin{cases} -(a+b \int _{\Omega } | \nabla u | ^2dx )\Delta u= \lambda u^3 + g(u) -h(x) & \mathrm{in}\,\, \Omega ,\\ u=0 & \mathrm{on}\,\, \partial \Omega,\end{cases}\tag{1} \] where \(\Omega \) is a smooth bounded domain in \({\mathbb R}^N\) \((N=1,2,3)\), \(a\geq 0\) and \(b>0\) are constants, \(h\in L^2(\Omega )\), and \(g \in C({\mathbb R},{\mathbb R})\) satisfies \[ \lim _{| t | \to \infty }\frac{g(t)}{t^3}=0. \leqno{(2)} \] The functional corresponding to problem (1) is \[ I_{\lambda }(u)= \frac{a}{2}\int _{\Omega } | \nabla u | ^2 dx + \frac{b}{4} \left( \int _{\Omega } | \nabla u | ^2 dx \right)^2-\frac{\lambda }{4}\int _{\Omega } u^4 dx - \int _{\Omega }G(u)dx + \int _{\Omega }hu dx \] defined on the Hilbert space \(X:=H_0^1(\Omega )\) where \(G(t)= \int _0^t g(s)ds\). Define \[ F(t) = \begin{cases} \frac{4}{3t} \int _0^t g(s)ds -\frac{1}{3} g(t), & t\neq 0, \\ g(0), & t=0, \end{cases} \] and let \[ \begin{aligned} \underline{F(+\infty )} = \liminf _{t\to + \infty }F(t), & \overline{F(+\infty )}= \limsup _{t\to + \infty }F(t), \\ \underline{F(-\infty )}= \liminf _{t\to - \infty }F(t),& \overline{F(-\infty )}= \limsup _{t\to - \infty }F(t). \end{aligned} \] Then the auhors get the following theorems. {Theorem 1.} Suppose that \(a>0\), \(g\in C({\mathbb R},{\mathbb R})\) satisfies (2), \(h\in L^2(\Omega )\) and \(\underline{F(-\infty )}>-\infty \), \(\overline{F(+ \infty )}< + \infty \). Then problem (1) has at least one weak solution in \(X\). {Theorem 2.} Suppose that \(a=0\), \(g\in C({\mathbb R},{\mathbb R})\) satisfies (2), \(h\in L^2(\Omega )\) and \(\underline{F(+\infty )}>-\infty \), \(\overline{F(- \infty )}< + \infty \). Assume that the following condition is satisfied \[ \int _{\Omega } hv dx < \underline{F(+\infty )}\int _{\Omega } v^+ dx - \overline{F(-\infty )} \int _{\Omega }v^-dx \] for all \(v \in A_{\lambda }\setminus \{0\}\) where \(A_{\lambda }\) is the set of eigenfunctions of some nonlinear eigenvalue problem for a given \(\lambda \). Then problem (1) has at least one weak solution in \(X\). {Theorem 3.} Suppose that \(a=0\), \(g\in C({\mathbb R},{\mathbb R})\) satisfies (2), \(h\in L^2(\Omega )\) and \(\underline{F(-\infty )}>-\infty \), \(\overline{F(+ \infty )}< + \infty \). Assume that the following condition is satisfied \[ \int _{\Omega } hv dx < \underline{F(-\infty )}\int _{\Omega } v^+ dx - \overline{F(+\infty )} \int _{\Omega }v^-dx \] for all \(v \in A_{\lambda }\setminus \{0\}\). Then problem (1) has at least one weak solution in \(X\).
scientific article; zbMATH DE number 6157296