One-dimensional nonlinear boundary value problems with variable exponent
The authors prove the existence of at least one non-zero solution of a nonlinear boundary value problem with variable exponent of the form \[ -(|u'(x)|^{p(x)-2}u'(x))'+a(x) |u(x)|^{p(x)-2} u(x)= \lambda f(x,u(x)), \quad u(0)=0=u(1), \] where $f$ is an $L^1$-Carathéodory function, $a\in L^{\infty}[0,1]$ with $\operatorname{essinf} a \geq 0, p \in C([0,1]$ with $\min p >1$ and $\lambda$ a positive real parameter. It is assumed a technical condition stated by means of an algebraic inequality for the primitive of $f$; no condition at zero or at infinity on the nonlinear term is required. The assumptions of the main result are satisfied e.g. when $p(x)=x^2, \lambda =1$ and $f(x,u)=x^4 u^2$. For the proof, it is applied an abstract result in Orlicz-Sobolev spaces due to the first author.
scientific article; zbMATH DE number 6810415