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Transactions of the American Mathematical Society

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

Authors: J. García Azorero and I. Peral Alonso
Journal: Trans. Amer. Math. Soc. 323 (1991), 877-895
MSC: Primary 35J65; Secondary 35B30
MathSciNet review: 1083144
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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}$

$\displaystyle - \operatorname{div} (\vert\nabla u{\vert^{p - 2}}\nabla u) = \ve... ...{\vert^{{p^{\ast}} - 2}}u + \lambda \vert u{\vert^{q - 2}}u,\qquad \lambda > 0,$

where $ {p^{\ast}}$ is the critical Sobolev exponent, and $ u{\vert _{\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.

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