Infinitely Many Solutions for a Fourth-Order Semilinear Elliptic Equations Perturbed from Symmetry

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Infinitely Many Solutions for a Fourth-Order Semilinear Elliptic Equations Perturbed from Symmetry Duong Trong Luyen1,2 Received: 18 August 2020 / Revised: 17 September 2020 / Accepted: 21 September 2020 © Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2020

Abstract In this paper, we study the existence of multiple solutions for the following biharmonic problem 2 u = f (x, u) + g(x, u) in , u = u = 0 on ∂, where  ⊂ R N , (N > 4) is a smooth bounded domain and f (x, ξ ) is odd in ξ, g(x, ξ ) is a perturbation term. By using the variant of Rabinowitz’s perturbation method, under some growth conditions on f and g, we show that there are infinitely many weak solutions to the problem. Keywords Biharmonic · Boundary value problems · Critical points · Perturbation methods · Multiple solutions Mathematics Subject Classification Primary 35J60; Secondary 35B33 · 35J25

1 Introduction In the last decades, the biharmonic elliptic equation 2 u = f (x, u) u = u = 0

in , on ∂,

(1.1)

Communicated by Maria Alessandra Ragusa.

B

Duong Trong Luyen [email protected]

1

Division of Computational Mathematics and Engineering, Institute for Computational Science, Ton Duc Thang University, Ho Chi Minh City, Vietnam

2

Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam

123

D. T. Luyen

has been studied by many authors see [4,6–8,16–18] and the references therein. In this paper, we study the existence of multiple weak solutions to the following problem 2 u = f (x, u) + g(x, u) u = u = 0

in , on ∂,

(1.2)

where  ⊂ R N , (N > 4) is a smooth bounded domain. To study the problem (1.2), we make the following assumptions: We assume that f :  × R → R is a function such that (A1) f (x, ξ ) = f 1 (x, ξ ) + f 2 (x, ξ ), f 1 , f 2 ∈ C( × R, R) and there exist constants C1 > 0 and 1 < p < 2 such that | f 1 (x, ξ )| ≤ C1 |ξ | p−1 , (x, ξ ) ∈  × R;

(1.3)

(A2) there exist constants C2 > 0 and 1 < μ < 2 such that f 1 (x, ξ )ξ − μF1 (x, ξ ) ≤ 0, (x, ξ ) ∈  × R, where F1 (x, ξ ) :=



f 1 (x, τ )dτ ;

0

(A3) there exist constants C2 > 0, 1 < p1 < 2 and 2 < p2 < 2∗ such that F1 (x, ξ ) ≥ C2 (|ξ | p1 − |ξ | p2 ), (x, ξ ) ∈ 0 × R, where 2∗ := N2N −4 , 0 is a nonempty open and 0 ⊂ ; (A4) there exist constants C3 > 0 and 2 < p3 < 2∗ such that | f 2 (x, ξ )| ≤ C3 |ξ | p3 −1 , (x, ξ ) ∈  × R; (A5) f i (x, ξ ) = − f i (x, −ξ ), i = 1, 2, (x, ξ ) ∈  × R. Let g :  × R → R is a function such that (B) g ∈ C( × R, R) and there exist constants C4 > 0 and 2 < θ < 2∗ such that |g(x, ξ )| ≤ C4 |ξ |θ−1 , (x, ξ ) ∈  × R. Now, we formulate the main result of this paper. Theorem 1.1 Assume that (A1)–(A5), (B) are satisfied and N 2p > . 2− p θ −2

(1.4)

Then the problem (1.2) has a sequence of small negative energy solutions converging to zero.

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Infinitely Many Solutions for a Fourth-Order Semilinear…

Example 1.2 Let  be a bounded domain with smooth boundary in R5 and 1

3

f (x, ξ ) = a(x) |ξ |− 2 ξ cos |ξ | 2 , g(x, ξ ) = ξ θ−1 , where a(x) ∈ C(

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