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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.

MSC:

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
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