Variable-exponent extension of the multiplicity and singular existence results

The paper considers the boundary value problems denoted by

$$and$$

, whose fixed-exponent formulations involve the $p$-Laplacian, the discontinuous Heaviside nonlinearity, and, in the singular case, the term $\lambda u^{-\gamma}$; the exponent $q$ is constant in the established results.

Variable-exponent conjecture. Theorems concerning multiplicity for

$$and existence for$$

should remain valid when $q$ is replaced by a variable exponent $q$.

This proposes extending the paper's principal results from constant-exponent Sobolev spaces to variable-exponent settings. The supplied text gives no resolution or further conditions for this extension.