Generalized quasilinear equations with critical growth and nonlinear boundary conditions

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