On a class of double-phase problem without Ambrosetti–Rabinowitz-type conditions

ABSTRACT

We study the following double-phase problem $\begin{array}{rl}& -\mathrm{d}\mathrm{i}\mathrm{v}(|\mathrm{\nabla }u{|}^{p-2}\mathrm{\nabla }u+a(x\left)|\mathrm{\nabla }u{|}^{q-2}\mathrm{\nabla }u\right)=\lambda f(x,u),\text{in}\mathrm{\Omega },\\ & u=0,\text{on}\mathrm{\partial }\mathrm{\Omega },\end{array}$ where $N\ge 2$ and 1<p<q<N. The aim is to determine the precise positive interval of λ for which the problem admits at least two nontrivial solutions via variational methods for the above equation. The primitive of the nonlinearity f is of super-q growth near infinity in u and allowed to be sign-changing. Furthermore, our assumptions are suitable and different from those studied previously.

Keywords:

• Double-phase problem
• variational method
• multiple solutions
• sign-changing potential

Mathematics Subject Classification (2010):

• 35J60
• 03H10
• 35D05

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work is supported by the National Natural Science Foundation of China (No. 11201095), the Youth Scholar Backbone Supporting Plan Project of Harbin Engineering University (No. 307201411008), the Fundamental Research Funds for the Central Universities (No. 2019), the Postdoctoral Research Startup Foundation of Heilongjiang (No. LBH-Q14044) and the Science Research Funds for Overseas Returned Chinese Scholars of Heilongjiang Province (No. LC201502).