Nonvariational and singular double phase problems for the Baouendi-Grushin operator

In the present paper, we consider the following fractional double phase problem with nonlocal reaction: \begin{align*} \qquad \quad \left\{ \begin{array}{lll} (- \Delta)_p^s u + (- \Delta)_q ^s u + V(\varepsilon x) (|u|^{p - 2} u + |u|^{q - 2} u) = \left(\frac{1}{|x|^\mu} * F(u)\right) f(u) & \ \text{in}\ \mathbb{R}^N, \\ u \in W^{s,p}(\mathbb{R}^N) \cap W^{s,q}(\mathbb{R}^N) ,\ u \gt 0 & \ \text{in}\ \mathbb{R}^N, \end{array} \right\} \end{align*} where ɛ is apositive parameter, $0 \lt s \lt 1$ , $2 \leqslant p \lt q \lt \min\{2p, N / s\}$ , $0 \lt \mu \lt sp$ , $(- \Delta)_t^s$ $(t \in \left\{p,q\right\})$ is the fractional t-Laplace operator, the reaction term $f : \mathbb{R} \mapsto \mathbb{R}$ is continuous, and the potential $V \in C (\mathbb{R}^N , \mathbb{R})$ satisfying a local condition. Using a variational approach and topological tools (the non-standard C1-Nehari manifold analysis and the abstract category theory), multiplicity of positive solutions and concentration properties for the above problem are established. Our results extend and complement some previous contributions related to double phase variational integrals.

show abstract

scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.

Contact Info

customersupport@researchsolutions.com

10624 S. Eastern Ave., Ste. A-614

Henderson, NV 89052, USA

This site is protected by reCAPTCHA and the Google Privacy Policy and Terms of Service apply.

Blog Terms and Conditions API Terms Privacy Policy Contact Cookie Preferences Do Not Sell or Share My Personal Information

Made with 💙 for researchers

Part of the Research Solutions Family.

Nonvariational and singular double phase problems for the Baouendi-Grushin operator

Cited by 4 publications

References 30 publications

A multiphase eigenvalue problem on a stratified Lie group

A multiphase eigenvalue problem on a stratified Lie group

Existence and nonexistence of positive radial solutions of a quasilinear Dirichlet problem with diffusion

Concentration and multiplicity of solutions for fractional double phase problems

Contact Info

Product

Resources

About