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A uniform $C^1$ connecting lemma
Deformation from symmetry and multiplicity of solutions in non-homogeneous problems
| 1. | Department of Mathematics, The University of British Columbia, Vancouver, B.C. V6M 3X7, Canada |
| 2. | Department of Mathematics, The University of British Columbia, Vancouver BC Canada V6T 1Z2, Canada |
$-\Delta u = p(x, u) + f(\theta, x, u)\quad $ on $\Omega$
$u = 0\quad$ on $\partial \Omega$
where $p(x, \cdot)$ is odd and $f$ is a perturbative term. An application of this result is the problem
$-\Delta u = \lambda |u|^{q-1}u + |u|^{p-1}u + f\quad$ on $\Omega$
$u = u_0\quad$ on $\partial \Omega$
where $\Omega$ is a smooth, bounded, open subset of $\mathbf R^n (n \geq 3), \lambda > 0, 1\leq q < p, f \in C(\bar \Omega, \mathbf R)$ and $u_0\in C^2(\partial \Omega, \mathbf R)$. It is proven that this equation has an infinite number of solutions for $p < \frac{n+1}{n-1}$ and that for any sub-critical $p$ i.e., $p < \frac{n+2}{n-2}$, there are as many solutions as we like, provided $||f||_{frac{p+1}{p}}$ and $||u_0||_{p+1}$ are small enough.
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