Multiplicity results for nonhomogeneous elliptic equations with singular nonlinearities

Abstract

This paper is concerned with the study of multiple positive solutions to the following elliptic problem involving a nonhomogeneous operator with nonstandard growth of p-q type and singular nonlinearitieswhere Ω is a bounded domain inLp,qu := div(|∇u| p−2 ∇u + |∇u| q−2 ∇u), 1 < p < q < ∞, γ ∈ (0, 1), and f is a continuous nondecreasing map satisfying suitable conditions. By constructing two distinctive pairs of strict sub and super solution, and using fixed point theorems by Amann [1], we prove existence of three positive solutions in the positive cone of C δ (Ω) and in a certain range of λ.

Strong comparison principle for a p-Laplace equation involving singularity and its applications

“…A strong comparison principle for p-q Laplace operator is proved in Proposition 7 of [11] which provides the Proposition 3.3 of our paper and that in turn can be used to prove a three solution theorem. Arora [3] has proved a result in this direction for the double phase operator. Though Remark 3.8 is an easy consequence of [3], we shall nevertheless conclude our paper showing the result for p Laplacian.…”

“…Arora [3] has proved a result in this direction for the double phase operator. Though Remark 3.8 is an easy consequence of [3], we shall nevertheless conclude our paper showing the result for p Laplacian. u δ ; u| ∂Ω = 0 for certain range of λ > 0 where p > 1, δ ∈ (0, 1) and Ω an arbitrary bounded domain in R N .…”

Combined effects in mixed local-nonlocal stationary problems

“…Moreover, using the integral representation via Green function and maximum principle, they proved the sharp boundary behavior of the weak solution. For further issues on nonlocal and nonlinear singular problems, the interested reader can consult to the bibliographic references in [8,6,21,23,35].…”