Generalized quasilinear equations with critical growth and nonlinear boundary conditions
We study the quasilinear problem $$\displaylines{ -\text{div}(h^2(u)\nabla u) + h(u)h'(u)|\nabla u|^2+u =-\lambda |u|^{q-2}u+|u|^{2 \cdot 2^*-2}u\quad \text{in } \Omega, \cr \frac{\partial u}{\partial\eta}= \mu g(x,u) \quad \text{on } \partial \Omega, }$$ where \(\Omega \subset \mathbb{R}^3\) i...
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| Published in: | Electronic journal of differential equations Vol. Special Issues; no. Special Issue 01; pp. 327 - 344 |
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| Main Authors: | , , |
| Format: | Journal Article |
| Language: | English |
| Published: |
Texas State University
27-06-2022
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| Subjects: | |
| Online Access: | Get full text |
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| Summary: | 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 |
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| ISSN: | 1072-6691 1072-6691 |
| DOI: | 10.58997/ejde.sp.01.m3 |