A topological analysis of p ( x )-harmonic functionals in one-dimensional nonlocal elliptic equations
Year: 2025
Type: article
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 ∫ 0 1 u ′ ( x ) p ( x ) d x u ′ ′ ( t ) = λ f t , 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 data u ( 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.
Source: Advanced Nonlinear Studies
Author: Christopher S. Goodrich
Institution: UNSW Sydney
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: gold
Estimated APC paid: $1,469
OpenAlex ID: W4409828463