The jumping nonlinearity problem revisited: an abstract approach

David G. Costa, Hossein Tehrani

DOI: http://dx.doi.org/10.12775/TMNA.2003.015

Abstract


We consider a class of nonlinear problems of the form
$$
Lu+g(x,u)=f,
$$
where
$L$ is an unbounded self-adjoint operator on a Hilbert space $H$
of $L^{2}(\Omega)$-functions, $\Omega\subset\mathbb{R}^{N}$ an arbitrary
domain, and $g\colon \Omega\times\mathbb{R}\rightarrow\mathbb{R}$ is
a ``jumping nonlinearity'' in the sense that the limits
$$
\lim_{s\rightarrow-\infty} \frac{g(x,s)}{s}=a
\quad\text{\rm and}\quad
\lim_{s\rightarrow\infty}\frac{g(x,s)}{s}=b
$$
exist and ``jump'' over an eigenvalue of the operator $-L$. Under
rather general conditions on the operator $L$ and for suitable $a< b$, we
show that a solution to our problem exists for any $f\in H$. Applications are
given to the beam equation, the wave equation, and elliptic equations in the
whole space $\mathbb{R}^{N}$.

Keywords


Jumping nonlinearity; critical point theory; beam equation; wave equation; Schrödinger equation

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