On some fourth-order semilinear elliptic problems in \(\mathbb R^N\). (English) Zbl 1011.35045

The purpose of this paper is to establish the existence of at least two solutions \(u\in D^{2,2}(\mathbb R^N)\setminus \{ 0\}\) to the problem \(\Delta^2u-\lambda g(x)u=f(x)|u|^{p-2}u\) in \(\mathbb R^N\), where \(N\geq 5\), \(\lambda >0\), \(p=2N/(N-4)\), and \(f\) changes sign in a prescribed manner. The proof of the main result is based on variational tools and it can be summarized as follows. The authors first examine Palais-Smale sequences in order to find the range of a variational functional associated to the above problem for which the Palais-Smale condition holds. Then it is applied the mountain-pass principle in order to obtain a first solution of the problem. Next, it is established the existence of a second solution, provided \(\lambda\) is in a small right neighbourhood of the first eigenvalue to the corresponding weighted eigenvalue problem. The key fact in the proof of the last assertion is that \(f(x)\) changes sign.
The paper gives an interesting perspective for the treatment of wide classes of fourth-order semilinear problems with lack of compactness.


35J30 Higher-order elliptic equations
31B30 Biharmonic and polyharmonic equations and functions in higher dimensions
49J35 Existence of solutions for minimax problems
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
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