Robin problems with general potential and double resonance
The authors study the Robin problem \[ \Delta u+\xi(z)u=f(z,u)\text{ in }\Omega, \frac{\partial u}{\partial n}+\beta(z)u=0\text{ on }\partial \Omega,\tag{P} \] where \(\Omega\) is a \(C^2\) bounded open domain in \(\mathbb{R}^N\), \(\xi\in L^s(\Omega)\) for some \(s>n\), \(\beta\in W^{1,\infty}(\partial \Omega)\), with \(\beta(z)\geq 0\) on \(\partial \Omega\), and \(f:\Omega\times \mathbb{R}\rightarrow \mathbb{R}\) is a Carathéodory function. The authors impose a summability condition on \(f(\cdot,x)\) and certain growth conditions at \(0\) and at infinity on \(f(z,\cdot)\), which include the linear growth condition \[ \hat{\lambda}_m\leq \liminf_{|x|\rightarrow \infty}\frac{f(z,x)}{x}\leq \limsup_{|x|\rightarrow \infty}\frac{f(z,x)}{x}\leq \hat{\lambda}_{m+1},\text{ uniformly in }z\in \Omega,\tag{1} \] where \(\hat{\lambda}_m,\hat{\lambda}_{m+1}\) are positive consecutive eigenvalues of the operator \(-\Delta +\xi(z)\). Note that condition \((1)\) allows a double resonance situation. Using variational methods and critical group theory, the authors establish an existence result of \(C^1(\overline{\Omega})\)-solutions to problem (P).
scientific article; zbMATH DE number 6697802
scientific article