Multiplicity of Nodal Solutions for a Class of Double- Phase Problems

<jats:p>We consider the following parametric double-phase problem: <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:mfenced open="{" close=""><mml:mrow><mml:mtable class="cases"><mml:mtr><mml:mtd columnalign="left"><mml:mo>−</mml:mo><mml:mi mathvariant="normal">div</mml:mi><mml:mfenced open="(" close=")"><mml:mrow><mml:msup><mml:mrow><mml:mfenced open="|" close="|"><mml:mrow><mml:mo>∇</mml:mo><mml:mi>u</mml:mi></mml:mrow></mml:mfenced></mml:mrow><mml:mrow><mml:mi>p</mml:mi><mml:mo>−</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo>∇</mml:mo><mml:mi>u</mml:mi><mml:mo>+</mml:mo><mml:mi>a</mml:mi><mml:mfenced open="(" close=")"><mml:mrow><mml:mi>x</mml:mi></mml:mrow></mml:mfenced><mml:msup><mml:mrow><mml:mfenced open="|" close="|"><mml:mrow><mml:mo>∇</mml:mo><mml:mi>u</mml:mi></mml:mrow></mml:mfenced></mml:mrow><mml:mrow><mml:mi>q</mml:mi><mml:mo>−</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo>∇</mml:mo><mml:mi>u</mml:mi></mml:mrow></mml:mfenced><mml:mo>=</mml:mo><mml:mi>λ</mml:mi><mml:mi>f</mml:mi><mml:mfenced open="(" close=")"><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>u</mml:mi></mml:mrow></mml:mfenced><mml:mtext> </mml:mtext><mml:mtext>in</mml:mtext><mml:mtext> </mml:mtext><mml:mi>Ω</mml:mi><mml:mo>,</mml:mo></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="left"><mml:mi>u</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mtext>on</mml:mtext><mml:mtext> </mml:mtext><mml:mi>∂</mml:mi><mml:mi>Ω</mml:mi><mml:mo>.</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced></mml:math>. We do not impose any global growth conditions to the nonlinearity <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M2"><mml:mi>f</mml:mi><mml:mfenced open="(" close=")"><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>u</mml:mi></mml:mrow></mml:mfenced></mml:math>, which refer solely to its behavior in a neighborhood of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M3"><mml:mi>u</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:math>. And we will show that they suffice for the multiplicity of signed and nodal solutions of the double-phase problem above when <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M4"><mml:mi>λ</mml:mi></mml:math> is large enough.</jats:p>


Introduction and Statement of Results
In this paper, we deal with the existence and multiplicity of solutions for the following double-phase problem: −div ∇u j j p−2 ∇u + a x ð Þ ∇u j j q−2 ∇u À Á = λf x, u ð Þ, in Ω, where λ > 0 is a parameter, Ω ⊂ ℝ N ðN ≥ 2Þ is a bounded domain with smooth boundary, 1 < p < q < min fN, p * g, p * = Np/ðN − pÞ, and we also assume that the nonlinearity f satisfies the following conditions: (f 1 ) f : Ω × ℝ ⟶ ℝ is a C 1 function, f ðx, 0Þ = 0 for a:e: x ∈ Ω (f 2 ) There exists γ ∈ ðq, p * Þ such that uniformly for a:e:x ∈ Ω (f 3 ) There exists β ∈ ðp, p + ðpðp * − pÞ/qÞÞ such that uniformly for a:e:x ∈ Ω, where Fðx, tÞ = Ð t 0 f ðx, sÞds (f 4 ) There exists a constant θ ∈ ðq, p * Þ, δ > 0 such that for a:e:x ∈ Ω and all 0 < jtj ≤ δ (f 5 ) For the δ in ðf 4 Þ, f ðx, −uÞ = −f ðx, uÞ, ∀x ∈ Ω, juj ≤ δ Similar problems have been investigated, and it is well known they have a strong physical meaning because they appear in the models of strongly anisotropic materials (see [1,2]). The energy functionals of the form where the integrand ℋ which switches between two different elliptic behaviors has been intensively studied since the late eighties (see [3][4][5][6][7][8][9]). Recently, Colombo and Mingione in [7] have obtained the regularity theory for minimizers of (6). The double-phase problem has been studied extensively recently. The existence of a signchanging ground state solution to problem (7) has been proven by Liu and Dai in [3], when f were assumed to satisfy the p-superlinear growth condition and Ambrosetti-Rabinowitz condition. In [4], by using Morse theory, Perera and Squassina obtained a nontrivial weak solution of problem (7), when f ðx, uÞ = λjuj p−2 u + juj r−2 u + hðx, uÞ. In [5], by utilizing the Nehari method, Liu and Dai obtained three ground state solutions. Motivated by the above works, we intend to establish the multiplicity of both signed and nodal solutions of problem (P λ ), when λ > 0 is large enough. Here, we note that the assumptions (f 1 )-(f 5 ) that we make on the nonlinearity f ðx, uÞ refer only to its behavior in a neighborhood of u = 0.
We present the main result of this paper as follows: Then, there exists Λ > 0, and if λ > Λ, problem (P λ ) has at least one positive solution, one negative solution, and a sign-changing solution.
Remark 3. From ðf 1 Þ and ð f 2 Þ, it can be seen that γ ≤ β. We may think of f ðx, tÞ = jtj β−2 t with θ < β, which clearly satisfy We remark that [3,5] obtained only one sign-changing solution. However, in Theorem 2, since k is arbitrary, we get infinitely many sign-changing solutions. To the best of our knowledge, little has been done in the literature on the existence of multiple nodal solutions for the parametric Dirichlet problem with the minimal conditions on the nonlinearity f ðx, uÞ.
The proof will be done by variational techniques. Since we have no information on the behavior of the nonlinearity f ðx, uÞ at the infinity, we adapt the argument introduced by Costa and Wang [10], which consists in making a suitable modification on f , solving a modified problem, and then checking that, for a large enough λ, the solutions of the modified problem are indeed solutions of the original one.
The paper is organized as follows. In Section 2, we modify the original problem and prove the Palais-Smale condition for the modified functional. We present some tools which are useful to establish a multiplicity result. In Section 3, we prove Theorem 1. Theorem 2 is proven in Section 4.

The Modified Problem
To prove our main results, we need to present the variational setting of our problem. Firstly, we introduce some notations and some necessary definitions. The Musielak-Orlicz space L ℋ ðΩÞ associated with the function consists of all measurable functions u : The Musielak-Orlicz space L ℋ ðΩÞ is defined by endowed with the norm The space L ℋ ðΩÞ is a separable, uniformly convex, and reflexive Banach space. We denote by k⋅k p the norm in L p ðΩÞ and by L q a ðΩÞ the space of all measurable functions u : Ω ⟶ ℝ with the seminorm It is easy to check that the embeddings The related Sobolev space W 1,ℋ ðΩÞ is defined by equipped with the norm Journal of Function Spaces where k∇uk ℋ = kj∇ujk ℋ . The completion of C ∞ 0 ðΩÞ in W 1,ℋ ðΩÞ is denoted by W 1,ℋ 0 ðΩÞ, and it can be equivalently renormed by via a Poincare-type inequality (cf [6], Proposition 2.18(iv)), under assumption (2). The spaces W 1,ℋ ðΩÞ and W 1,ℋ 0 ðΩÞ are uniformly convex and hence are reflexive, Banach spaces. By Proposition 2.15 in [6], we know that the Sobolev embedding W 1,H 0 ðΩÞ↪L r ðΩÞ is compact since r < p * . By (14), We have From now on, we denote by X ≔ W 1,ℋ 0 ðΩÞ for convenience in writing. We for a:e:x ∈ Ω and all jtj ≤ δ. Now, define the even cutoff function ζðtÞ ∈ C 2 ðℝ, ½0, 1Þ as Moreover, for all t ∈ ℝ, ζðtÞ satisfies that LetF Also, we setf From hypotheses ðf 1 Þ − ð f 4 Þ, it is easy to check thatf is a Carathéodory function and satisfies the following properties. (ii) There exists μ = min fθ, γg such that for a:e:x ∈ Ω and t ≠ 0 Proof. The proof is similar to that of Lemma 1.1 in [10]. Now let us consider the modified problem of (P λ ): The corresponding energy functional of (P * λ ) is We easily get that the functionalĨ λ ðuÞ ∈ C 2 ðX, ℝÞ, and its critical points are the solutions of (P * λ ). We note that solutions of (P * λ ) with kuk ∞ ≤ δ are also solutions of (P λ ). We shall search solutions of (P * λ ) as critical points ofĨ λ ðuÞ. Let us now define Jð⋅Þ: X ⟶ ℝ as and we denote the derivative operator by A, that is, In the following lemma, we summarize some properties of A, useful to study our problem.

Lemma 5.
Under the condition (2), A is a mapping of type (S + ); that is, if u n ⇀ u in X and lim sup n→+∞ hAðu n Þ − AðuÞ, u n − ui ≤ 0, then u n ⟶ u in X.
Proof. The proof is similar to that of Proposition 3.1(ii) in [3].
Firstly, we show the functionalĨ λ satisfies the (PS) condition.

Journal of Function Spaces
We claim that fu n g is bounded in X. Indeed, if ku n k ≤ 1, we have done. If ku n k > 1, by Lemma 4(ii), then we have that where μ = min fθ, γg > q. Hence, fu n g is bounded. Therefore, there is a subsequence (which we still denote by fu n g) that converges weakly to some u ∈ X and strongly in L γ ðΩÞ.
It is easy to check from Lemma 4(i) and Hölder's inequality that Then, So u n ⟶ u follows from Lemma 5. Next, we will show the functionalĨ λ satisfies the Mountain Pass Geometry [11].

Proof of Theorem 1
In this section, we prove our main result. We will show (P * λ ) has a positive solution, a negative solution, and a nodal solution. And the solutions obtained satisfy the estimate kuk ∞ ≤ δ. This fact implies that these solutions are indeed solutions of the original problem (P λ ).
Similarly, we can definẽ whereF − ðx, uÞ = Ð u 0f − ðx, sÞds: We also get a negative solution u − λ < 0 of our original problem (P λ ) for all λ ≥ λ 3 ≥ λ 2 . We next show that there is a sign-changing solution for λ large enough. We can apply the method introduced by Li and Wang [14] to our case. Since X is a real, reflexive, and separable Banach space, there are fe j g ⊂ X and fe * j g ⊂ X * such that X = span e j , j = 1, 2,⋯ È É , X * = span e * j , j = 1, 2,⋯ n o , For k = 1, 2, ⋯, we denote Y k ≔ span e 1 , e 2 , ⋯, e k f g , Z k ≔ span e k , e k+1 , ⋯ f g : ð60Þ On X, we define a closed convex cone P X = u ∈ Xju x ð Þ ≥ 0, a:e:in Ω f g :