Existence of infinitely many weak solutions for a Neumann problem
The paper is concerned with the Neumann problem \[ -\Delta_p u= \mu f(x,u)+h(x,u), \quad \text{in } \Omega, \] \[ \partial u/\partial\nu=0 \quad \text{on } \partial\Omega, \] on a bounded \(C^1\)-domain \(\Omega\subset\mathbb R^N\). Here \(\Delta_p\) is the \(p\)-Laplace operator, \(p>1\), \(\mu\) is a parameter, and \(f,h:\Omega\times\mathbb R\to\mathbb R\) are nonlinear functions. The function \(f(x,t)\) is of order \(O(| t|^{p-1})\) as \(| t|\to\infty\), and \(h(x,t)=O(| t|^q)\) for some \(q<(p-1)N+p(N-p)\). The main theorem states that under an additional condition on \(H(x,t)=\int^t_0 h(x,s)\,ds\), there exists \(\mu^*\in(0,\infty]\) such that \((P_\mu)\) for \(|\mu|<\mu^*\) has an unbounded sequence of solution in \(W^{1,p}(\Omega)\). As an example, the theorem applies with \(\mu^*=\infty\) if \(H(x,t)=-\lambda(x)| t|^m g(\text{{ln}}| t|)\) with \(\lambda\in L^\infty(\mathbb R)\), \(\text{{essinf}}\,\lambda>0\), \(p<m 0\). The proof uses variational methods and is based on a critical point theorem of \textit{B. Ricceri} [J. Comput. Appl. Math. 113, 401--410 (2000; Zbl 0946.49001)].
scientific article; zbMATH DE number 2102567
scientific article