### The jumping nonlinearity problem revisited: an abstract approach

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 $\mathbb{R}^{N}$.

$$

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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