The Existence of Nontrivial Solutions to a Class of Quasilinear Equations

<jats:p>In this paper, we study the following quasilinear equation: <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M1">
                        <mo>−</mo>
                        <mi mathvariant="normal">div</mi>
                        <mfenced open="(" close=")">
                           <mrow>
                              <mi>ϕ</mi>
                              <mfenced open="(" close=")">
                                 <mrow>
                                    <mfenced open="|" close="|">
                                       <mrow>
                                          <mo>∇</mo>
                                          <mi>u</mi>
                                       </mrow>
                                    </mfenced>
                                 </mrow>
                              </mfenced>
                              <mo>∇</mo>
                              <mi>u</mi>
                           </mrow>
                        </mfenced>
                        <mo>+</mo>
                        <mi>ϕ</mi>
                        <mfenced open="(" close=")">
                           <mrow>
                              <mfenced open="|" close="|">
                                 <mrow>
                                    <mi>u</mi>
                                 </mrow>
                              </mfenced>
                           </mrow>
                        </mfenced>
                        <mi>u</mi>
                        <mo>=</mo>
                        <mi>f</mi>
                        <mfenced open="(" close=")">
                           <mrow>
                              <mi>u</mi>
                           </mrow>
                        </mfenced>
                        <mtext> </mtext>
                        <mtext>in</mtext>
                        <mtext> </mtext>
                        <msup>
                           <mrow>
                              <mi>ℝ</mi>
                           </mrow>
                           <mrow>
                              <mi>N</mi>
                           </mrow>
                        </msup>
                     </math>
                  </jats:inline-formula>, where <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M2">
                        <mi>ϕ</mi>
                        <mo>∈</mo>
                        <msup>
                           <mrow>
                              <mi>C</mi>
                           </mrow>
                           <mrow>
                              <mn>1</mn>
                           </mrow>
                        </msup>
                        <mfenced open="(" close=")">
                           <mrow>
                              <msup>
                                 <mrow>
                                    <mi>ℝ</mi>
                                 </mrow>
                                 <mrow>
                                    <mo>+</mo>
                                 </mrow>
                              </msup>
                              <mo>,</mo>
                              <msup>
                                 <mrow>
                                    <mi>ℝ</mi>
                                 </mrow>
                                 <mrow>
                                    <mo>+</mo>
                                 </mrow>
                              </msup>
                           </mrow>
                        </mfenced>
                     </math>
                  </jats:inline-formula> and <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M3">
                        <mi>Φ</mi>
                        <mfenced open="(" close=")">
                           <mrow>
                              <mi>t</mi>
                           </mrow>
                        </mfenced>
                        <mo>=</mo>
                        <msubsup>
                           <mrow>
                              <mo>∫</mo>
                           </mrow>
                           <mrow>
                              <mn>0</mn>
                           </mrow>
                           <mrow>
                              <mi>t</mi>
                           </mrow>
                        </msubsup>
                        <mi>s</mi>
                        <mi>ϕ</mi>
                        <mfenced open="(" close=")">
                           <mrow>
                              <mo>∣</mo>
                              <mi>s</mi>
                              <mo>∣</mo>
                           </mrow>
                        </mfenced>
                        <mi>d</mi>
                        <mi>s</mi>
                     </math>
                  </jats:inline-formula>. In the Orlicz-Sobolev space, by variational methods and a minimax theorem, we prove the equation has a nontrivial solution.</jats:p>


Introduction
In this paper, we consider the following quasilinear elliptic equation: where N ≥ 3 is the dimension of space and ϕ ∈ C 1 ðℝ + , ℝ + Þ satisfies the following conditions: (ϕ 1 ) lim t⟶0 ϕðtÞt = 0, lim t⟶+∞ ϕðtÞt = +∞, ðϕðtÞtÞ ′ > 0, for t > 0 (ϕ 2 ) There exist p, q ∈ ð1, NÞ, q < p < q * ≡ Nq/ðN − qÞ, such that q − 2 ≤ ðϕ ′ ðtÞtÞ/ðϕðtÞÞ ≤ p − 2, for t > 0 Φ is a continuous function on ℝ defined by and Φ-Laplace is defined by Let The quasilinear elliptic equation is an important partial differential equation; in recent years, many researchers have studied the following equation: where Ω ⊂ ℝ N is an open set, 1 < p < N. Under some growth conditions, many people proved the existence and multiplicity of solutions to (5) and several mathematicians also obtained the bifurcation and asymptotic properties. By variational method and maximal principle, Guo [1] and Guo-Webb [2] obtained the existence and uniqueness of the solution to (5) and they also considered the partial symmetric properties of the solutions. Guedda-Veron [3] used topology and spectrum analysis to study the bifurcation and multiplicity of the solutions. We point out that by constructing the pseudogradient vectors of p-Laplace operator, Zhang-Li [4] firstly obtained the sign-changing solutions to (5) and see also Zhang-Chen-Li [5], for more results of p-Laplace equations, one can see [6][7][8] and the references therein. By variational method and minimax theorem, Li-Guo [9] and Li-Liang [10] studied the p − q-Laplace equation and they obtained the existence and multiple solutions.
In [11], Franchi-Lanconelli-Serrin studied the quasilinear equation (1), and in weighted Sobolev space, they considered the existence and uniqueness of the solution to (1). In fact, the function ϕ is a general elliptic equation. For example, if ϕðuÞ ≡ 1, then (1) is the Laplace equation; if ϕðuÞ = u p−2 , then (1) is the p-Laplace equation; ϕðuÞ = 1/ ffiffiffiffiffiffiffiffiffiffiffi ffi 1 + p 2 p , then (1) is the curvature equation. The Orlicz-Sobolev space is a kind of general norm space and one can study the quasilinear equations in this space. Carvalho-Silva-Goulart [12] and Carvalho-Silva-Goncalves-Goulart [13] considered (1) in the Orlicz-Sobolev space, by variational method and concentration-compactness theorem. They obtained the existence of (1), and they also studied the famous problem, i.e., concave and convex terms.
In this paper, let f ðtÞ satisfy the following conditions: Under the preceding conditions, by a variational method, we obtain the existence of a nontrivial solution to (1). We follow the idea in [10] to obtain the existence of nontrivial solution. Our conditions ð f 1 Þ-ð f 5 Þ are slightly different from what is in [10]. Conditions ðf 1 Þ and ð f 2 Þ are standard. Our condition ð f 3 Þ is the improvement of ð f 3 Þ in [10], which is used to obtain the superlinear growth of f at t ⟶ +∞.
The condition ð f 4 Þ is to guarantee that A is a Finsler manifold which is used to obtain a special minimizing sequence (see . Condition ð f 5 Þ is the compactness condition. Remark 2. By ð f 2 Þ and (7), we have that This paper is organized as follows. In Section 2, we recall some results of the Orlicz-Sobolev space; in Section 3, we list and prove some preliminary results; and in Section 4, we prove our main theorem.

Orlicz-Sobolev Spaces
In this section, we recall some useful knowledge for the Orlicz space and Orlicz-Sobolev space and give some inequalities on Φ. The reader can refer to [14,15] for more details.
By condition ðϕ 1 Þ and the definition of Φ, the function Φ is a N-function (see [14] for the definition of N-function).
For any Ω ⊂ ℝ N , under the assumptions ðϕ 1 Þ and ðϕ 2 Þ, the Orlicz space L Φ ðΩÞ contains all measurable functions u : and the Luxemburg norm on L Φ ðΩÞ is defined by For any integer m ≥ 1, the corresponding Orlicz-Sobolev space W m L Φ ðΩÞ is defined by and the norm on W m L Φ ðΩÞ is defined by Let E Φ ðΩÞ denote the closure in L Φ ðΩÞ of function u which are bounded on Ω and have bounded support in Ω.
In the following, for simplicity, we denote kuk Φ,ℝ N and kuk m,Φ,ℝ N by kuk Φ and kuk m,Φ . Then, the Orlicz-Sobolev space has the following properties.
Theorem 9 (Theorem 8.12 [14]). The imbedding L B ðℝ N Þ ⊂ L A ðℝ N Þ holds if and only if B dominates A globally, i.e., there exists a constant k > 0 such that AðtÞ ≤ BðktÞ holds for all t ≥ 0.
Theorem 10 (Theorem 8.36 [14]). Let Ω be an arbitrary domain in ℝ N . If the N-function ΦðtÞ satisfies then W m 0 L Φ ðΩÞ ⊂ L Φ * ðΩÞ, for any integer m ≥ 1. Moreover, if Ω 0 is a bounded subdomain of Ω, then the imbedding W m 0 L Φ ðΩÞ ⊂ L B ðΩ 0 Þ exists and is compact for any N-function B increasing essentially more slowly than Φ * near infinity.

Preliminary Results
In this section, we prove some preliminary results for future use.

Journal of Function Spaces
To study the existence of solution to (1), we first study its energy functional. It is clear that the functional defining on W 1 L Φ ðℝ N Þ is given by It is easy to see that under the assumptions The h·, · i is the dual pairing between W 1 L Φ ðℝ N Þ and its dual space By condition ð f 2 Þ and ð f 3 Þ, we have that for any ε > 0, there exists a constant C ε > 0 such that Hence, it follows that for any ε > 0, there exists a constant C ε > 0 such that Lemma 11. Suppose ð f 5 Þ holds. Then, there exists a positive constant C > 0 such that for any u ∈ ℝ + , t ∈ ℝ + , Proof. By ðf 5 Þ, one has that, for s ∈ ℝ + , Integrating the last inequality from u to tu, we get the result in the lemma. This ends the proof. Set Then, we show that A is not an empty set.
Proof. By ð f 1 Þ and ðf 2 Þ, f ≡0. Hence, there exists a constant a 0 > 0 such that It is easy to show that hðtÞ is continuous. For t > 1, by (7), ð f 5 Þ, and Lemma 11, we have Using the definition of u 0 , one has that For 0 < t < 1, by (7), we have By (9), we can see It follows that, for any 0 < ε < 1, there exists somet > 0 such that, for any t ∈ ð0,tÞ, Hence, Therefore, there exists some t 0 > 0 such that hðt 0 Þ = 0. This ends the proof.
It is easy to see that H is a C 1 -functional. For any u ∈ A, one has that u + ≡0. If u + ≡ 0, then f ðuÞ ≡ 0 by ð f 1 Þ. Noting that u≡0, one has that HðuÞ > 0, which contradicts with u ∈ A. By ðϕ 2 Þ and ðf 4 Þ

Journal of Function Spaces
Hence, A is a closed and a complete submanifold of W 1 L Φ ðℝ N Þ with the natural Finsler structure (see [16]). Using Lemma 2.14 [10] with n = 1, e 1 = u/hG ′ ðuÞ, ui, one has the following.

Lemma 15. For any
with π is the projection from W 1 L Φ ðℝ N Þ to T u A.
By Lemma 12, A is not empty, and by Lemma 14, I is bounded from below. Hence, by Lemma 2.15 of [10] and Lemma 15, we can get the following result.
Lemma 17. Let fρ n g ⊂ L 1 ðℝ N Þ be a bounded sequence and ρ n ≥ 0, then there exists a subsequence, still denoted by fρ n g, such that one of the following two possibilities occurs: (2) (Nonvanishing): there exists α > 0, 0 < R < +∞ and fy n g ⊂ ℝ N , such that lim n⟶∞ Ð y n +B R ρ n ðxÞdx ≥ α > 0 Lemma 18. Suppose fu n g is a bounded sequence in W 1 L Φ ðℝ N Þ, and for some r > 0, Then, lim n⟶∞ Ð ℝ N gðju n jÞju n jdx = 0: Proof. By Lemma 7, (7), and Hölder inequality, for any τ ∈ ð0, 1Þ, Noting that fu n g is bounded in W 1 L Φ ðℝ N Þ, there exists a constant M > 0 such that Then, it follows from imbedding theorem (25) and Lemma 4 that for some positive constants C 1 and C 2 . Thus, by Lemma 5, ð for some positive constants C 3 and C 4 . Letting 1 − τ = p/q * , it follows from (58) and (61) that Journal of Function Spaces for some C > 0. Then, coving ℝ N by balls of radius r, in such a way that each point of ℝ N is contained in at most N + 1 balls, we have as n ⟶ ∞. Noting that fu n g is bounded in as n ⟶ ∞. By imbedding theorem (25) and Lemma 7, one sees that Ð ℝ N Φ * ′ð|u n | Þ | u n | dx is a bounded sequence. If λ < τ, setting μ = ðτ − λÞ/τ, then as n ⟶ ∞. This ends the proof.
Theorem 19 (Vitali's convergence theorem [17]). Let ðX, A, μÞ be a measure space and let 1 ≤ p < ∞. Let f f n g ∞ n=1 be a sequence in L p ðX, A, μÞ and let f be an A -measurable function such that f is finite μ-a.e. and f n ⟶ f μ-a.e. Then, f ∈ L p ðX, A, μÞ and k f − f n k p ⟶ 0 if and only if (i) For each ε > 0, there exists a set A ε ∈ A such that μðA ε Þ < ∞ and Ð A c ε jf n j p dμ < ε for all n ∈ ℝ N (ii) lim μðEÞ⟶0 Ð E j f n j p dμ = 0 uniformly in n, i.e., for each ε > 0, there is a δ > 0 such that E ∈ A and μðEÞ < δ imply Ð E j f n j p dμ < ε for all n ∈ ℕ.
Proof. By Lemma 16, there exists a sequence fu n g ⊂ A, such that Step 1. There exists some constant C > 0 such that ku n k 1,Φ ≤ C: By (66) and (7), one has In the last step, we use condition (f 5 ). Hence, by Lemma 4, there exists some constant C > 0 such that ku n k 1,Φ ≤ C: Step 2. There exist R > 0, α > 0, and fy n g ⊂ ℝ N , such that Notice u n ∈ A, we obtain Journal of Function Spaces This is a contradiction. Hence, only nonvanishing is possible, i.e., there exist R > 0, α > 0, and fy n g ⊂ ℝ N , such that Step 3. There exists u ∈ W 1 L Φ ðℝ N Þ, such that ∇ u n ⟶ ∇ u a.e. in ℝ N , where u n ðxÞ = u n ðx + y n Þ.
For any ψ ∈ W 1 L Φ ðℝ N Þ, set ψ n ðxÞ = ψðx − y n Þ. Then, kψk 1,Φ = kψ n k 1,Φ , and as n ⟶ ∞. Since k u n k 1,Φ = k u n k 1,Φ , so f u n g is a bounded sequence in W 1 L Φ ðℝ N Þ. Hence, there exists u ∈ W 1 L Φ ðℝ N Þ and u≡0 such that some sequence of f u n g, still denoted by f u n g, u n ⇀ u in W 1 L Φ ℝ N À Á , u n ⟶ u a:e:in ℝ N , ð Ω g u n − u j j ð Þ u n − u j jdx ⟶ 0 for any bounded subset Ω ⊂ ℝ N : Let η r ∈ C ∞ 0 ðℝ N Þ such that 0 ≤ η ≤ 1, with η r = 1 in B r = fx ∈ ℝ N | jxj ≤ rg, η r = 0 in B c 2r , and j∇η r j ≤ C/r for some C > 0. Then, it is easy to see that On the other hand, By (17), (26), and Lemma 4, Noting that u n is bounded in W 1 L Φ ðℝ N Þ, for any ε > 0, there exists r 0 > 0 such that for r > r 0 ,