Generalized Picone inequalities and their applications to $(p,q)$-Laplace equations

Abstract

We obtain a generalization of the Picone inequality which, in combination with the classical Picone inequality, appears to be useful for problems with the $(p,q)$-Laplace type operators. With its help, as well as with the help of several other known generalized Picone inequalities, we provide some nontrivial facts on the existence and nonexistence of positive solutions to the zero Dirichlet problem for the equation $-{\Delta }_{p}u-{\Delta }_{q}u=f_{\mu }(x,u,\nabla u)$ in a bounded domain $\Omega \subset {\mathbb{R}}^N$ under certain assumptions on the nonlinearity and with a special attention to the resonance case $f_{\mu }(x,u,\nabla u)={\lambda }_1(p)|u{|}^{p-2}u+\mu |u{|}^{q-2}u$, where ${\lambda }_1(p)$ is the first eigenvalue of the $p$-Laplacian.

arXiv comment18 pages, 1 figure. Remark 1.3 added, formulation and proof of Lemma 1.6 slightly improved, figure added, inequality (1.12) added, several minor changes according to referee's suggestions incorporated

References in corpus (3)

• On positive solutions for $(p,q)$-Laplace equations with two parameters
• Uniqueness for unbounded solutions to stationary viscous Hamilton--Jacobi equations
• Symmetry and stability of non-negative solutions to degenerate elliptic equations in a ball

Cited by in corpus (5)

• On generalized eigenvalue problems of fractional $(p,q)$-Laplace operator with two parameters
• On the antimaximum principle for the $p$-Laplacian and its sublinear perturbations
• Multiplicity of positive solutions for $(p,q)$-Laplace equations with two parameters
• On subhomogeneous indefinite $p$-Laplace equations in supercritical spectral interval
• On the linear independence condition for the Bobkov-Tanaka first eigenvalue of the double-phase operator