On some generalizations of the Landesman-Lazer theorem. (English) Zbl 1127.47056

For a bounded domain \(\Omega\subseteq{\mathbb R}^N\) with smooth boundary, and for \(p\geq2\) such that \(2p>N\), consider a linear (unbounded) Fredholm operator \(A\) in \(L^p(\Omega)\) with domain \(D(A)=W^{2,p}(\Omega)\cap W^{1,p}_0(\Omega)\). Here \(W^{k,p}(\Omega)\) is the Sobolev space of functions with \(k\) distributional derivatives in \(L^p(\Omega)\), and \(W^{1,p}_0(\Omega)\) is the closure of \(C_{\text{c}}^\infty(\Omega)\) in \(W^{1,p}(\Omega)\). Suppose that \(A\) is symmetric with respect to the scalar product in \(L^2(\Omega)\), and that \(\ker A\) is the 1-dimensional space spanned by a function \(\omega\in D(A)\setminus\{0\}\).
Let \(g(r)\) be a given continuous function on \({\mathbb R}\) with existing limits \(g(\pm\infty)\) as \(r\to\pm\infty\), such that \(g(-\infty)\leq g(r)\leq g(+\infty)\) for all \(r\). Suppose that \(\phi : \Omega\times{\mathbb R}\to{\mathbb R}\) has existing limits \[ \underline{\phi}(x,\xi):=\liminf_{\xi'\to\xi}\phi(x,\xi') \qquad\text{and}\qquad \overline{\phi}(x,\xi):=\limsup_{\xi'\to\xi}\phi(x,\xi') \] for almost all \(x\in\Omega\), and that \(\underline{\phi},\overline{\phi}\) are superpositionally measurable. Finally, suppose that there are \(f_*,f^*\in L^p(\Omega)\) such that \(f_*(x)\leq\phi(x,\xi)\leq f^*(x)\) for a.e. \(x\in\Omega\) and all \(\xi\in{\mathbb R}\).
Under the Landesman–Lazer type conditions \[ \begin{aligned} \int_{\omega>0} f^*\omega\,dx+\int_{\omega<0} f_*\omega\,dx &<g(+\infty)\int_{\omega>0}\omega\,dx+g(-\infty)\int_{\omega<0}\omega\,dx,\\ \int_{\omega>0} f_*\omega\,dx+\int_{\omega<0} f^*\omega\,dx &>g(-\infty)\int_{\omega>0}\omega\,dx+g(+\infty)\int_{\omega<0}\omega\,dx, \end{aligned} \] it is proved that the equation \[ (Au)(x)+g(u(x))=\phi(x,u(x)) \] has a generalized solution. The same result is also proved for nonsymmetric \(A\) with higher dimensional kernel, albeit under abstract conditions in the functional setting.
The authors present an example involving a resonant nonlinearly oscillating membrane with a discontinuous obstacle, and and example on Lavrentiev’s problem on detachable currents at the presence of resonance.
The proofs involve an application of coincidende index theory for multivalued maps.


47N20 Applications of operator theory to differential and integral equations
47J05 Equations involving nonlinear operators (general)
35R05 PDEs with low regular coefficients and/or low regular data
47H10 Fixed-point theorems
47H04 Set-valued operators