Existence of a ray of eigenvalues for equations with discontinuous operators (Q2773672)

The authors consider the question of existence of eigenfunctions of the following nonlinear problem: NEWLINE\[NEWLINE Au=\lambda Tu,\quad u\in E,\;\lambda>0, \tag{1}NEWLINE\]NEWLINE where \(E\) is a real reflexive Banach space, the symbol \(E^*\) stands for its adjoint, \(A\:E\to E^*\) is a selfadjoint operator, and \(T\:E\to E^*\) denotes a compact mapping bounded on \(E\) with \(T(0)=0\). The mapping \(T\) is not assumed to be continuous. The main conditions on \(T\) and \(A\) can be stated as follows:NEWLINENEWLINENEWLINEa) there exists a functional \(f\:E\to\mathbb R\) such that NEWLINE\[NEWLINE f(x+h)-f(x)= \int_0^1 (T(x+th),h) dt\quad (x,h\in E), NEWLINE\]NEWLINE where the symbol \( (\cdot,\cdot)\) denotes the duality between \(E\) and \(E^*\);NEWLINENEWLINENEWLINEb) \(\lim_{t\to +0}(T(x+th)-Tx,h)\geq 0\) for all \(x,h\in E\);NEWLINENEWLINENEWLINEc) \(E=E_1+E_2\), with \(E_1=\text{ker} A\), and \((Au,u)\geq \delta\|u\|_E^2\) (\(\delta>0\)) for every \(u\in E_2\);NEWLINENEWLINENEWLINEd) \(f(0)=0\) and there exists \(u_0\in E\) such that \(f(u_0)>0\); moreover, \(\lim_{u\in E_1, \|u\|_E\to \infty} f(u)=-\infty\) whenever \(E_1\neq \{0\}\).NEWLINENEWLINENEWLINEUnder the above conditions, it is proven that there exists \(\lambda_0>0\) such that, for every \(\lambda>\lambda_0\), there exists a nontrivial solution to problem (1). This solution is a minimizer of the functional \(f^\lambda (u)=\frac{1}{2}(Au,u)-\lambda f(u)\) and a point of radial continuity of \(T\), i.e., NEWLINE\[NEWLINE \lim_{t\to 0}(T(u+th),h)=(Tu,h),\quad h\in E. NEWLINE\]NEWLINE The results are applied to the study of boundary value problems for the elliptic operator NEWLINE\[NEWLINE Lu(x)=-\sum_{i,j=1}^n (a_{ij}u_{x_i})_{x_j} + c(x)u(x)=\lambda g(x,u(x)),\quad x\in \Omega. NEWLINE\]NEWLINE The bounded measurable function \(g\) is not continuous. It is assumed that, for almost every \(x\in \Omega\), NEWLINE\[NEWLINE g(x,u)\in [g _{-}(x,u),g_{+}(x,u)],NEWLINE\]NEWLINE NEWLINE\[NEWLINEg_{-}(x,u)=\varliminf_{\eta\to u}g(x,\eta),\quad g_{+}(x,u) =\varlimsup_{\eta\to u} g(x,\eta),\quad g(x,0)=0, NEWLINE\]NEWLINE where \(g_{\pm}\) are bounded functions.