On positive stable solutions to weighted quasilinear problems with negative exponent

Liouville‐type theorem for nonlinear elliptic equations involving p‐Laplace–type Grushin operators

“…If γ =0, p =2 and b = θ =0, then q c (2, N ,0,0)= p 0 , which is the critical exponent p 0 in and equals to the exponent in Ma and Wei . Moreover, Theorem recovers the known result for the p ‐Laplace operator in Le et al, theorem 1.1 when γ =0. Therefore, our conclusions in Theorems and extend some results in the above references.…”

“…After that, the result in Wang and Ye was extended to problem with f ( u )= e u in Huang et al and equation −△ p u = f ( x ) g ( u ) in Chen et al, where g ( u )= e u or g ( u )=− u − q . Similar works on singular problems can be founded in other studies …”

“…Proof We apply the idea of test functions in Le et al to discuss the above statement. As mentioned in Lemma , the solution is not assumed to be locally bounded, u τ ψ is not, a priori, a suitable test function for any τ <0, even with Then, we need to construct a sequence of suitable cut‐off functions.…”

Liouville theorems for stable weak solutions of elliptic problems involving Grushin operator

“…where q > 0. Such generalizations may be found in [6,17,19,20] and references therein. Some attempts to extend Farina's results to elliptic problems with nonlinearity f belonging to a wider class of positive and convex functions were also made in [4,10,16,18].…”

“…The usual cut-off functions as used in [9,23] do not work with general nonlinearity f . To overcome this difficulty, we use new cut-off functions inspired by [17,19,20] and show that our key estimates are still valid for these functions. • We also construct some examples to show the sharpness of our Liouville theorems.…”