Rui He, Sihua Liang, Existence results for the double-phase (p(x), q(x))-Laplace equation involving convolution terms and critical frequency

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DOI: 10.23952/jnva.8.2024.5.08

Volume 8, Issue 5, 1 October 2024, Pages 813-828

Abstract. In this paper, we investigate the double-phase $(p(x), q(x))$ -Laplace equation involving a convolution term and critical frequency of the form:
$\mathcal{L}(u) +V(x)(|u|^{p(x)-2}u+|u|^{q(x)-2}u)= \lambda\left(\int_{\mathbb{R}^N}\frac{F(y,u(y))}{|x-y|^{\alpha(x,y)}}dy \right)f(x,u)+|u|^{p^{\ast}(x)-2}u$ in $\mathbb{R}^N,$
where $\mathcal{L}(u) = -\text{div}(|\nabla u|^{p(x)-2}\nabla u)-\text{div}(|\nabla u|^{q(x)-2}\nabla u)$ , $V$ and $f$ are continuous functions, and $p^\ast(x)=Np(x)/(N-p(x))$ is the critical Sobolev exponent for variable exponents. By using the variational methods and some analytical skills, we obtain the existence and multiplicity of nontrivial solutions for the above problem.

How to Cite this Article:
R. He, S. Liang, Existence results for the double-phase (p(x), q(x))-Laplace equation involving convolution terms and critical frequency, J. Nonlinear Var. Anal. 8 (2024), 813-528.