On resonant Neumann problems: multiplicity of solutions
The authors consider the following semilinear Neumann problem \[ \begin{cases} -\Delta u = f(z,u(z)) & \text{in } \Omega ,\\ \frac{\partial u }{\partial n} =0 & \text{in }\Omega, \end{cases}\tag{1} \] where \(\Omega \subset {\mathbb R}^N\) is a bounded domain with \(C^2\) boundary \(\partial \Omega \). Let \(\{ \lambda _m\}_{m=1}^{\infty }\) be the distinct eigenvalues of \((-\Delta ,H^1(\Omega ))\). Assumptions on the function \(f(z,x)\) \((z \in \Omega ,x \in {\mathbb R}^N)\) are as follows: {\parindent=8mm \begin{itemize}\item[(i)] for all \(x \in {\mathbb R}\), \(z \to f(z,x)\) is measurable; \item[(ii)] for almost all \(z \in \Omega \), \(x \to f(z,x)\) is \(C^1\) and \(f(z,0)=0\); \item[(iii)] for almost all \(z \in \Omega \), and all \(x \in {\mathbb R}\), \(| f_x'(z,x)| \leq a(z)+ c| x | ^{r-2}\) with \(a \in L^{\infty }(\Omega )\), \(c>0\) and \(2<r<2^*\) where \(2^*= 2N/(N-2)\) if \(N\geq 3\) and \(+ \infty \) if \(N\in \{1,2\}\); \item[(iv)] there exists an integer \(m\geq 1\) such that \[ \lim _{| x | \to \infty } \frac{f(z,x)}{x} = \lambda _m \quad \text{uniformly for a.e. } z\in \Omega ; \] \item[(v)] there exist functions \(\xi ,\widehat{\xi }\in L^{\infty }(\Omega )_+\) such that \(\xi (z)\leq \lambda _m\) a.e. on \(\Omega \), the inequality is strict on a set of positive measure and \[ -\widehat{\xi }(z)(x-x')^2 \leq (f(z,x)- f(z,x'))(x-x')\leq \xi (z) (x-x')^2 \] for a.e. \( z\in \Omega \), and \( x,x' \in {\mathbb R}\); \item[(vi)] if \(F(z,x)= \int _0^x f(z,s)ds\), then \(\lim _{| x | \to \infty }(f(z,x)x-2F(z,x)) = -\infty \) for a.e. \(z \in \Omega \); \item[(vii)] there exists a function \(\eta \in L^{\infty }(\Omega )\) such that \(\eta (z) \leq 0\) a.e. on \(\Omega \), the inequality is strict on a set of positive measure and \[ \lim _{x \to 0} \frac{2F(z,x)}{x^2} \leq \eta (z) \] uniformly for a.e. \(z \in \Omega \). \end{itemize}} Here (iv) and (vi) imply that (1) is resonant at infinity with respect to \(\lambda _m \, (m\geq 1)\) from the right. Under the hypotheses (i)--(vii), the authors prove that the problem (1) has at least four nontrivial solutions. In the proof, variational methods are used, based on the critical point theory, reduction theory and Morse theory.
scientific article; zbMATH DE number 5959706
scientific article