Infinitely many solutions for some singular elliptic problems
Andrzej Szulkin Shoyeb Waliullah
We prove the existence of an unbounded sequence of critical points of the functional \begin{equation*} J_{\lambda}(u) =\frac{1} {p} ∫_{\mathbb{R} ^N}{||x|^{-α\nabla^k} u|} ^p - λ h(x){||x|^{-α+k}u|} ^p - \frac{1} {q} ∫_{\mathbb{R} ^N}Q(x){||x|^{-b}u|} ^q \end{equation*} related to the Caffarelli-Kohn-Nirenberg inequality and its higher order variant by Lin. We assume $Q\le 0$ at 0 and infinity and consider two essentially different cases: $h\equiv 1$ and $h$ in a certain weighted Lebesgue space.
keywords: infinitely many solutions. Nehari manifold sign-changing weight function Caffarelli-Kohn-Nirenberg inequality
p-Laplacian problems where the nonlinearity crosses an eigenvalue
Kanishka Perera Andrzej Szulkin
Using linking arguments and a cohomological index theory we obtain nontrivial solutions of $p$-Laplacian problems with nonlinearities that interact with the spectrum.
keywords: linking. p-Laplacian cohomological index

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