Existence of multiple positive weak solutions and estimates for extremal values for a class of concave-convex elliptic problems with an inverse-square potential

In this paper, variational methods are used to establish some existence and multiplicity results and provide uniform estimates of extremal values for a class of elliptic equations of the form: \begin{document}$-Δ u - {{λ}\over{|x|^2}}u = h(x) u^q + μ W(x) u^p,\ \ x∈Ω\backslash\{0\}$ \end{document} with Dirichlet boundary conditions, where \begin{document}$0∈ Ω\subset\mathbb{R}^N $\end{document} ( \begin{document}$N≥q 3 $\end{document} ) be a bounded domain with smooth boundary \begin{document}$\partial Ω $\end{document} , \begin{document}$μ>0 $\end{document} is a parameter, \begin{document}$0 , \begin{document}$h(x)>0 $\end{document} and \begin{document}$W(x) $\end{document} is a given function with the set \begin{document}$\{x∈ Ω: W(x)>0\} $\end{document} of positive measure.

Still few general results are known. In the case λ = 0, R(u) = (1 + u) p (that is, −∆u = µ(1 + u) p ) and 1 < p ≤ 2 * − 1, F. Gazzola and A. Malchiodi [8, pp.811, Theorem 1] proved that there exists a constant µ 1 > 0 such that the problem has at least two positive solutions u µ and U µ , where u µ is minimal (in the sense that u µ (x) ≤ v µ (x) for all x ∈ Ω and for any other solution v µ of (P λ,µ )) and U µ is a mountain-pass solution when µ ∈ (0, µ 1 ), has a unique solution U * in H 1 0 (Ω) when µ = µ 1 and has no solution when µ > µ 1 even in distributional sense. In the case λ = 0, 0 < q < 1 < p = 2 * −1 and h, W ≡ 1, Y. Sun and S. Li [10, pp.1858, Theorem 1 and Theorem 2] proved that there exists a constant µ 2 > 0 such that the problem has at least two positive weak solutions if 0 < µ < µ 2 and has no solution if µ > µ 2 . In the case λ = 0 and p, q, h, W are under the same assumptions as in (1.1), Y. Sun [9, pp.752, Theorem 1.1] proved that there exists a constant µ 3 > 0 such that the problem has at least two positive weak solutions if 0 < µ < µ 3 and has no solution if µ > µ 3 . Thus, provided 0 < λ < Λ, it is natural to ask what the case would be for problem (1.1). Our goal of this paper is to show how variational methods can be used to establish some existence and multiplicity results for problem (1.1).
On H 1 0 (Ω), we use the norm Thanks to the Hardy inequality, the norm · λ is equivalent to the usual norm · of H 1 0 (Ω). Problem (1.1) is variational in nature, so for u ∈ H 1 0 (Ω), we define I λ,µ : the energy functional associated to problem (1.1). It is well known that there exists one-to-one correspondence between the weak solutions of problem (1.1) and the critical points of I λ,µ on H 1 0 (Ω). More precisely, we say that u ∈ H 1 0 (Ω) is a positive weak solution of problem (1.1) we mean a function u ∈ H 1 0 (Ω) such that u ≥ 0, u ≡ 0 (by the strong maximum principle then u > 0 in Ω\{0}) and for any ϕ ∈ H 1 0 (Ω) there holds A standard regularity argument shows that u ∈ C 2 (Ω\{0}), in which case, we say that u satisfies (1.1) in the classical sense. Of course, if such a solution exists, it must lie in the following Nehari type sets: In order to motivate our results, denote J λ,µ (u) = I λ,µ (u), u and decompose F λ,µ with F + λ,µ , F 0 λ,µ , F − λ,µ defined as follows: Our main results are as follows: First, inspired by [9] and [11], we take full advantage of the connection between the Nehari manifolds and the fibrering maps (that is, maps of the form t → I λ,µ (tu); see [3] and [7] for related results) to establish some existence and multiplicity results for problem (1.1): Then, as a by-product of the proof of Theorem 1.1, we obtain the blow up behavior of solution U λ,ε ∈ F − λ,µ of problem (1.1) with p = 1 + ε as ε → 0 + , like claimed in [8,pp.830,Theorem 15] for a non-singular problem and [9, pp.752, Theorem 1.3] for problem (1.1) provided λ = 0: Namely, U λ,ε blows up faster than exponentially with respect to ε.
Lastly, inspired by [1], we use the method of sub-and supersolutions to show that 1534 YAOPING CHEN AND JIANQING CHEN Theorem 1.3. There exists µ * = µ * (N, Ω, λ, q, p) > 0 such that the following problem (problem (1.1) when h, W ≡ 1) has at least a positive solution for every 0 < µ < µ * and has no solution for any µ > µ * . Remark 1. When λ = Λ, the operator −∆ − λ |x| 2 is no longer coercive in H 1 0 (Ω). However, we can still make use of the improved Hardy inequality (see [6, pp. to define a new Hilbert space, in which the operator is coercive, even when λ = Λ. We conjecture that it is a problem to prove the existence of solutions for problem (1.1) provided λ = Λ for further study, although there are still some difficulties.
This paper is organized as follows. Section 2 contains some notations and preliminaries. Section 3 is devoted to the proof of Theorem 1.1. In Section 4, we prove Theorem 1.2. The proof of Theorem 1.3 is contained in Sections 5. In Section 6, we give an appendix, where some proofs are shown.
This completes the proof of Lemma 5.1.
This completes the proof of Theorem 1.3.

Remark 2.
For µ = µ * , let {µ n } be a sequence such that µ n ↑ µ * and u n = u µn be a solution of problem (1.3) µn , that is, for any ϕ ∈ H 1 0 (Ω), there holds If one can show that {u n } is bounded in H 1 0 (Ω) (there are still some difficulties), then going if necessary to subsequence, we can assume that u n u * in H 1 0 (Ω), u n → u * a.e. in Ω, u n → u * in L 1+q (Ω) and L p+1 (Ω).
Letting n → ∞ from (14), we get that such a u * is thus a weak solution of problem (1.3) µ * , that is, for µ = µ * , problem (1.3) µ has at least a solution.

Proof. One can show that
where we use the relations (p − 1)(1 + q) This completes the proof of Appendix (2).