A topological analysis of p(x)-harmonic functionals in one-dimensional nonlocal elliptic equations

Abstract

We consider a class of one-dimensional elliptic equations possessing a p(x)-harmonic functional as a nonlocal coefficient. As part of our results we treat the model case−M∫01u′(x)p(x)dxu′′(t)=λft,u(t), 0<t<1 $${-}M\left(\underset{0{}}{\overset{1}{\int }}{\left\vert {u}^{\prime }(x)\right\vert }^{p(x)}\ \mathrm{d}x\right){u}^{\prime \prime }(t)=\lambda f\left(t,u(t)\right)\text{,\hspace{0.17em}}0< t< 1$$ subject to the boundary datau(0)=0=u(1). $$u(0)=0=u(1).$$ In addition, we consider a broader class of problems, of which the model case in a special case, by writing the argument of M as a finite convolution. As part of the analysis, a simple but fundamental lemma in introduced that allows the estimation of u′(x)p(x) ${\left\vert {u}^{\prime }(x)\right\vert }^{p(x)}$ in terms of constant exponents; this is the key to circumventing the variable exponent. An unusual array of analytical tools is used, including Sobolev’s inequality. Our results address both existence and nonexistence of solution.

Keywords

• nonlocal differential equation
• p(x)-harmonic functional
• one-dimensional sobolev’s inequality
• positive solution
• convolution
• primary: 33b15
• 34b10
• 34b18
• 42a85
• 44a35
• secondary: 26a33
• 46e35
• 47h30