Generalized quasilinear equations with critical growth and nonlinear boundary conditions

Abstract

We study the quasilinear problem $$\displaylines{ -\text{div}(h^2(u)\nabla u) + h(u)h'(u)|\nabla u|^2+u =-\lambda |u|^{q-2}u+|u|^{2 \cdot 2^*-2}u\quad \text{in } \Omega, \cr \frac{\partial u}{\partial\eta}= \mu g(x,u) \quad \text{on } \partial \Omega, }$$ where \(\Omega \subset \mathbb{R}^3\) is a bounded domain with regular boundary \(\partial \Omega\), \(\lambda,\mu>0\), \(1<q<4\), \(2\cdot2^{\ast}=12\), \(\frac{\partial }{\partial\eta}\) is the outer normal derivative and \(g\) has a subcritical growth in the sense of the trace Sobolev embedding. We prove a regularity result for all weak solutions for a modified, and introducing a new type of constraint, we obtain a multiplicity of solutions, including the existence of a ground state. For more information see  https://ejde.math.txstate.edu/special/01/m3/abstr.html