## 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