| Popis: |
Abstract This paper is concerned with the following double-phase problem: { − div ( | ∇ u | p − 2 ∇ u + a ( x ) | ∇ u | q − 2 ∇ u ) = f ( x , u ) in Ω , u = 0 on ∂ Ω , $$ \textstyle\begin{cases} -\operatorname{div}( \vert \nabla u \vert ^{p-2}\nabla u+a(x) \vert \nabla u \vert ^{q-2}\nabla u)=f(x,u)& \text{in }\varOmega, \\ u=0 &\text{on }\partial\varOmega, \end{cases} $$ where N ≥ 2 $N\geq2$ and 1 < p < q < N $1< p< q< N$ . Assuming that the primitive of f ( x , u ) $f(x,u)$ is asymptotically q-linear as | u | → ∞ $|u|\rightarrow\infty$ and a weak version of Nehari-type monotonicity condition that the function u ↦ f ( x , u ) | u | q − 1 $u\mapsto\frac{f(x,u)}{|u|^{q-1}}$ is nondecreasing on ( − ∞ , 0 ) ∪ ( 0 , ∞ ) $(-\infty,0) \cup(0,\infty)$ for a.e. x ∈ Ω $x\in\varOmega$ , we prove the existence of one ground state sign-changing solution via the constraint variational method and quantitative deformation lemma for the equation. Our results improve and generalize some results obtained by Liu and Dai (J. Differ. Equ. 265(9):4311–4334, 2018). |
