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- Title
Multiplicity of solutions to the weighted critical quasilinear problems.
- Authors
Liang, Sihua; Zhang, Jihui
- Abstract
We consider a class of critical quasilinear problems\begin{gather*}-\Div(|x|^{-ap}|\nabla u|^{p-2}\nabla u)-\mu\frac{|u|^{p-2}u}{|x|^{p(a+1)}}=\frac{|u|^{q-2}u}{|x|^{bq}}+\lambdaf(x,u)\quad\text{in~}\sOm,\\u=0\quad\text{on~}\partial\sOm,\end{gather*}where 0 ∈ Ω ⊂ ℝN, N ≥ 3, is a bounded domain and 1 < p < N, a < N/p, a ≤ b < a + 1, λ is a positive parameter, 0 ≤ μ < $\bar{\mu}$ ≡ ((N − p)/p − a)p, q = q*(a, b) ≡ Np/[N − pd] and d ≡ a+1 − b. Infinitely many small solutions are obtained by using a version of the symmetric Mountain Pass Theorem and a variant of the concentration-compactness principle. We deal with a problem that extends some results involving singularities not only in the nonlinearities but also in the operator.
- Subjects
MULTIPLICITY (Mathematics); CRITICAL point theory; QUASILINEARIZATION; MATHEMATICAL symmetry; MOUNTAIN pass theorem; MATHEMATICAL singularities; OPERATOR theory; NONLINEAR theories; DEGENERATE differential equations
- Publication
Proceedings of the Edinburgh Mathematical Society, 2012, Vol 55, Issue 1, p181
- ISSN
0013-0915
- Publication type
Article
- DOI
10.1017/S0013091509001813