A bifurcation result of Böhme-Marino type for quasilinear elliptic equations
The paper deals with the eigenvalue problem \[ \begin{cases}(\lambda,u)\in\mathbb{R}\times (H_0^1(\Omega)\cap L^{\infty}(\Omega)),\\ \int_{\Omega}\sum_{i,j=1}^na_{ij}(x,u)D_iuD_jv\,dx+\frac{1}{2}\int_{\Omega} \sum_{i,j=1}^nD_sa_{ij}(x,u)D_iuD_ju \,v\,dx\\ -\int_{\Omega}g(x,u)v\,dx=\lambda\int_{\Omega}uv\,dx,\,\,\,\text{for all}\,\,v\in H_0^1(\Omega)\cap L^{\infty}(\Omega),\end{cases}\leqno(P) \] where \(\Omega\) is a bounded open subset of \(\mathbb{R}^n\), and \(g,\,a_{ij}:\Omega\times \mathbb{R}\to\mathbb{R},\,\,i,j=\overline{1,n}\) satisfy suitable assumptions, among which \(g(x,0)=0\) for a.e. \(x\in\Omega\). Using some theorems from the nonsmooth critical point theory, the authors prove that \(\mu\in\mathbb{R}\) is a bifurcation value of problem \((P)\) if and only if \(\mu\) is an eigenvalue of the operator \(A:H_1^0(\Omega)\to H^{-1}(\Omega)\), defined by \[ \langle Au,v\rangle=\int_{\Omega}\sum_{i,j=1}^na_{ij}(x,0)D_iuD_jv\,dx-\int_{\Omega} D_sg(x,0)uv\,dx. \] In addition, they show the existence of two branches of bifurcation of Böhme-Marino type, from the trivial branch of solutions \(\{(\lambda,0);\,\,\lambda\in\mathbb{R}\}\) of \((P)\).
scientific article; zbMATH DE number 5310039