In this article, we show the existence of multiple positive solutions to a class of degenerate elliptic equations involving critical cone Sobolev exponent and sign-changing weight function on singular manifolds with the help of category theory and the Nehari manifold method.

Construction of Gauss quadratures with prescribed knots via Jacobi matrices is extended to the case where not all orthogonal polynomials exist due to the weight function changing sign. An algorithm is described and is demonstrated by calculating the knots of Kronrod schemes and other Gauss quadratures with prescribed knots.

We consider the elliptic problem $$ - \Delta_A u + = a_{\lambda}(x) |u|^{q-2}u+b_{\mu}(x) |u|^{p-2}u , for \(x \in \mathbb{R}^N\), \( 1 < q 2 p 2^*= 2N/(N-2)\), \(a_{\lambda}(x)\) is a sign-changing weight function, \(b_{\mu}(x)\) satisfies some additional conditions, \(u H^1_A(\mathbb{R}^N)\) and \(A:\mathbb{R}^N \to \mathbb{R}^N\) magnetic potential. Exploring Bahri-Li argument preliminary...

In this paper we study the existence, nonexistence and multiplicity of non-negative solutions for the family of problems −∆u = λ (a(x)e + f(x, u)), u ∈ H 0 (Ω) where Ω is a bounded domain in R2 and λ > 0 is a parameter. The coefficient a(x) is allowed to change sign. The techniques used in the proofs are a combination of upper and lower solutions, the TrudingerMoser inequality and variational m...