Existence of nontrivial solutions for generalized quasilinear Schrödinger equations with critical growth
Summary: We study the following generalized quasilinear Schrödinger equations with critical growth \(-\operatorname{div}(g^2 (u) \nabla u)+g(u) g' (u)|\nabla u |^2 +V (x) u = \lambda f(x,u)+g(u)| G(u)|^{2^\ast -2} G(u)\), \(x \in \mathbb R^N\), where \(\lambda >0\), \(N \geq 3\), \(g(s) : \mathbb R \rightarrow \mathbb R^+\) is a \(C^1\) even function, \(g(0)=1\), and \(g'(s)\geq 0\) for all \( \geq 0\), where \(G(u):=\int_0^u g(t) d t\). Under some suitable conditions, we prove that the equation has a nontrivial solution by variational method.
scientific article; zbMATH DE number 6991951
scientific article