The nodal solution for a problem involving the logarithmic and exponential nonlinearities

ABSTRACT

In this work, we study the existence of the least energy sign-changing solution for the following degenerate Kirchhoff-type problem involving the fractional N/s-Laplacian with logarithmic and both subcritical and critical exponential nonlinearities: $\{\begin{array}{l}{\displaystyle M\left({\int }_{{\mathbb{R}}^{2N}}\frac{|u(x)-u(y){|}^{\frac{N}{s}}}{|x-y{|}^{2N}}\mathrm{d}x\mathrm{d}y\right)(-\mathrm{\Delta }{)}_{N/s}^{s}u}\\ =|u{|}^{q-2}\mathrm{uln}|u{|}^2+\mathrm{\mu f}(u),& \mathrm{in}\mathrm{\Omega },\\ u=0,& \mathrm{in}{\mathbb{R}}^N\setminus \mathrm{\Omega },\end{array}$ where $\mathrm{\Omega }\subset {\mathbb{R}}^N$ is a bounded domain. The proof is based on constrained variational method, fractional Trudinger–Moser inequality, quantitative deformation lemma and Brouwer's degree theory in Nehari sets. To be more precise, the least energy sign-changing solution is obtained by minimizing the energy functional on the sign-changing Nehari manifold.

KEYWORDS:

• Fractional N/s-Laplacian
• Kirchhoff problem
• logarithmic nonlinearity
• exponential growth
• sign-changing solution

AMS SUBJECT CLASSIFICATIONS:

• Primary 35J35
• Secondary 35J60
• 35R11
• 35B33

Disclosure statement

No potential conflict of interest was reported by the author(s).

Acknowledgment

The authors are grateful to the referees for their useful suggestions which have improved the writing of the paper.

Additional information

Funding

Project supported by Natural Science Foundation of China [grant numbers 11501252, 11571176].