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On singular double phase problem with variable exponents and convolution term
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DOI：10.1515/math-2025-0228.png)
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Abstract
En 中文
In this article, we consider the following singular double phase problem with variable exponents and convolution term of the form: - T p ( x ) , q ( x ) alpha ( x ) ( u ) = lambda s ( x ) u - gamma ( x ) + integral Omega | u ( x ) | h ( x ) | x - y | mu ( x , y ) | u ( y ) | h ( x ) - 2 u ( y ) in Omega , u = 0 on partial derivative Omega , where the operator T p ( x ) , q ( x ) alpha ( x ) ( u ) : = div del u p ( x ) - 2 del u + alpha ( x ) del u q ( x ) - 2 del u is the double phase operator with variable exponents, Omega subset of R N is a bounded domain with smooth boundary partial derivative Omega, 0 <= alpha(& sdot;) is an element of L infinity(Omega), lambda is a positive real parameter. The functions s ( x ) is an element of C ( Omega ) are positive with compact support in Omega, h : R N -> R and mu : R N & times; R N -> R are continuous functions. Under the suitable conditions, the existence of at least one weak solution is obtained for the above problem by using the Nehari manifold approach. The novelty of this paper is that this problem includes singular term and convolution term. Moreover, the emergence of p(x) and q(x) Laplacian operator makes the study of this problem more complicated and interesting.
Keywords:
double phase problem
variable exponents
singular term
convolution term
Nehari manifold approach