Inbo Sim, Byungjae Son, On positive solutions for a class of double phase problems with strong singular weights and nonlinearities

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DOI: 10.23952/jnva.10.2026.1.03

Volume 10, Issue 1, 1 April 2026, Pages 61-85

Abstract. We study the existence and nonexistence of positive solutions for the singular double phase problem:
$-\left(\alpha(t)\varphi_p(u')+\beta(t)\varphi_q(u')\right)'= \lambda h(t) f(u),\ t\in(0,1),$

$u(0)=0=u(1),$
where $\lambda>0$ , $1\textless p\textless q\textless \infty$ , and $\varphi_m(s):=|s|^{m-2}s$ . This model can contain strong singular weights and nonlinearities such as $h(t)=t^{-\delta}$ with $\delta\geq 1$ and $f(u)=1+u^{-\gamma}$ with $\gamma\geq 1.$ Firstly, we provide sufficient conditions on $\alpha$ , $\beta$ , and $h$ such that positive solutions belong to $C[0,1]$ , which generalizes previous results. Secondly, we establish various existence results, including the existence of three positive solutions, which are new results even for strong singular $p$ -Laplacian problems. We prove the existence results by applying approximation techniques, resulting in approximation solutions that are not in $C^1[0,1]$ , stemming from the degeneracy of $\alpha$ and $\beta$ .

How to Cite this Article:
I. Sim, B. Son, On positive solutions for a class of double phase problems with strong singular weights and nonlinearities, J. Nonlinear Var. Anal. 10 (2026), 61-85.