## Multiplicity of solutions for elliptic problems with critical exponent or with a nonsymmetric term

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- by J. García Azorero and I. Peral Alonso
- Trans. Amer. Math. Soc.
**323**(1991), 877-895 - DOI: https://doi.org/10.1090/S0002-9947-1991-1083144-2
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## Abstract:

We study the existence of solutions for the following nonlinear degenerate elliptic problems in a bounded domain $\Omega \subset {{\mathbf {R}}^N}$ \[ - \operatorname {div} (|\nabla u{|^{p - 2}}\nabla u) = |u{|^{{p^{\ast }} - 2}}u + \lambda |u{|^{q - 2}}u,\qquad \lambda > 0,\] where ${p^{\ast }}$ is the critical Sobolev exponent, and $u{|_{\delta \Omega }} \equiv 0$. By using critical point methods we obtain the existence of solutions in the following cases: If $p < q < {p^{\ast }}$, there exists ${\lambda _0} > 0$ such that for all $\lambda > {\lambda _0}$ there exists a nontrivial solution. If $\max (p,{p^{\ast }} - p/(p - 1)) < q < {p^{\ast }}$, there exists nontrivial solution for all $\lambda > 0$. If $1 < q < p$ there exists ${\lambda _1}$ such that, for $0 < \lambda < {\lambda _1}$, there exist infinitely many solutions. Finally, we obtain a multiplicity result in a noncritical problem when the associated functional is not symmetric.## References

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## Bibliographic Information

- © Copyright 1991 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**323**(1991), 877-895 - MSC: Primary 35J65; Secondary 35B30
- DOI: https://doi.org/10.1090/S0002-9947-1991-1083144-2
- MathSciNet review: 1083144