## 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.

### MSC:

 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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### References:

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