×

Multiple solutions to a Cafarelli-Kohn-Nirenberg type equation with asymptotically linear term. (English) Zbl 1112.35074

Summary: We study the existence of multiple solutions to a Caffarelli-Kohn-Nirenberg type equation with asymptotically linear term at infinity
\[ \begin{cases} -\text{div}(| x| ^ {-\alpha p}| Du| ^ {p-2}Du)= | x| ^ {-(\alpha+1)p+c}f(u)&\text{ in }\;\Omega,\\ u=0&\text{ on }\;\partial\Omega,\end{cases} \]
where \(\Omega\subset{\mathbb R}^ {n}\) is a bounded regular domain such that \(0\in\Omega\), \(1<p<n\), \(0\leq\alpha<{n-p\over p}\), \(c>0\), and \(f\in C({\mathbb R})\) is such that \(f(t)=-f(-t)\) for all \(t\in{\mathbb R}\) and satisfies:
(f1) \(\lim_ {t\to 0}{f(t)\over | t| ^ {p-2}t}=0\),
(f2) \(\lim_ {t\to +\infty}{f(t)\over | t| ^ {p-2}t}=l<+\infty\).
In this case, the well-known Ambrosetti-Rabinowitz type condition doesn’t hold, hence it is difficult to verify the classical (PS)\(_c\) condition. To overcome this difficulty, we use an equivalent version of Cerami’s condition, which allows the more general existence result.

MSC:

35J60 Nonlinear elliptic equations
PDFBibTeX XMLCite
Full Text: arXiv EuDML