Existence and asymptotical behavior of solutions for a quasilinear Choquard equation with singularity


               <jats:p>In this study, we consider the following quasilinear Choquard equation with singularity <jats:disp-formula id="j_math-2021-0025_eq_001">
                     <jats:alternatives>
                        <jats:graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_math-2021-0025_eq_001.png" />
                        <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block">
                           <m:mfenced open="{" close="">
                              <m:mrow>
                                 <m:mtable displaystyle="true">
                                    <m:mtr>
                                       <m:mtd columnalign="left">
                                          <m:mo>−</m:mo>
                                          <m:mi mathvariant="normal">Δ</m:mi>
                                          <m:mi>u</m:mi>
                                          <m:mo>+</m:mo>
                                          <m:mi>V</m:mi>
                                          <m:mrow>
                                             <m:mo>(</m:mo>
                                             <m:mrow>
                                                <m:mi>x</m:mi>
                                             </m:mrow>
                                             <m:mo>)</m:mo>
                                          </m:mrow>
                                          <m:mi>u</m:mi>
                                          <m:mo>−</m:mo>
                                          <m:mi>u</m:mi>
                                          <m:mi mathvariant="normal">Δ</m:mi>
                                          <m:msup>
                                             <m:mrow>
                                                <m:mi>u</m:mi>
                                             </m:mrow>
                                             <m:mrow>
                                                <m:mn>2</m:mn>
                                             </m:mrow>
                                          </m:msup>
                                          <m:mo>+</m:mo>
                                          <m:mi>λ</m:mi>
                                          <m:mrow>
                                             <m:mo>(</m:mo>
                                             <m:mrow>
                                                <m:msub>
                                                   <m:mrow>
                                                      <m:mi>I</m:mi>
                                                   </m:mrow>
                                                   <m:mrow>
                                                      <m:mi>α</m:mi>
                                                   </m:mrow>
                                                </m:msub>
                                                <m:mo>∗</m:mo>
                                                <m:mo>∣</m:mo>
                                                <m:mi>u</m:mi>
                                                <m:msup>
                                                   <m:mrow>
                                                      <m:mo>∣</m:mo>
                                                   </m:mrow>
                                                   <m:mrow>
                                                      <m:mi>p</m:mi>
                                                   </m:mrow>
                                                </m:msup>
                                             </m:mrow>
                                             <m:mo>)</m:mo>
                                          </m:mrow>
                                          <m:mo>∣</m:mo>
                                          <m:mi>u</m:mi>
                                          <m:msup>
                                             <m:mrow>
                                                <m:mo>∣</m:mo>
                                             </m:mrow>
                                             <m:mrow>
                                                <m:mi>p</m:mi>
                                                <m:mo>−</m:mo>
                                                <m:mn>2</m:mn>
                                             </m:mrow>
                                          </m:msup>
                                          <m:mi>u</m:mi>
                                          <m:mo>=</m:mo>
                                          <m:mi>K</m:mi>
                                          <m:mrow>
                                             <m:mo>(</m:mo>
                                             <m:mrow>
                                                <m:mi>x</m:mi>
                                             </m:mrow>
                                             <m:mo>)</m:mo>
                                          </m:mrow>
                                          <m:msup>
                                             <m:mrow>
                                                <m:mi>u</m:mi>
                                             </m:mrow>
                                             <m:mrow>
                                                <m:mo>−</m:mo>
                                                <m:mi>γ</m:mi>
                                             </m:mrow>
                                          </m:msup>
                                          <m:mo>,</m:mo>
                                          <m:mspace width="1.0em" />
                                       </m:mtd>
                                       <m:mtd columnalign="left">
                                          <m:mi>x</m:mi>
                                          <m:mo>∈</m:mo>
                                          <m:msup>
                                             <m:mrow>
                                                <m:mi mathvariant="double-struck">R</m:mi>
                                             </m:mrow>
                                             <m:mrow>
                                                <m:mi>N</m:mi>
                                             </m:mrow>
                                          </m:msup>
                                          <m:mo>,</m:mo>
                                       </m:mtd>
                                    </m:mtr>
                                    <m:mtr>
                                       <m:mtd columnalign="left">
                                          <m:mi>u</m:mi>
                                          <m:mo>></m:mo>
                                          <m:mn>0</m:mn>
                                          <m:mo>,</m:mo>
                                          <m:mspace width="1.0em" />
                                       </m:mtd>
                                       <m:mtd columnalign="left">
                                          <m:mi>x</m:mi>
                                          <m:mo>∈</m:mo>
                                          <m:msup>
                                             <m:mrow>
                                                <m:mi mathvariant="double-struck">R</m:mi>
                                             </m:mrow>
                                             <m:mrow>
                                                <m:mi>N</m:mi>
                                             </m:mrow>
                                          </m:msup>
                                          <m:mo>,</m:mo>
                                       </m:mtd>
                                    </m:mtr>
                                 </m:mtable>
                              </m:mrow>
                           </m:mfenced>
                        </m:math>
                        <jats:tex-math>\left\{\begin{array}{ll}-\Delta u+V\left(x)u-u\Delta {u}^{2}+\lambda \left({I}_{\alpha }\ast | u{| }^{p})| u{| }^{p-2}u=K\left(x){u}^{-\gamma },\hspace{1.0em}& x\in {{\mathbb{R}}}^{N},\\ u\gt 0,\hspace{1.0em}& x\in {{\mathbb{R}}}^{N},\end{array}\right.</jats:tex-math>
                     </jats:alternatives>
                  </jats:disp-formula> where <jats:inline-formula>
                     <jats:alternatives>
                        <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_math-2021-0025_eq_002.png" />
                        <m:math xmlns:m="http://www.w3.org/1998/Math/MathML">
                           <m:msub>
                              <m:mrow>
                                 <m:mi>I</m:mi>
                              </m:mrow>
                              <m:mrow>
                                 <m:mi>α</m:mi>
                              </m:mrow>
                           </m:msub>
                        </m:math>
                        <jats:tex-math>{I}_{\alpha }</jats:tex-math>
                     </jats:alternatives>
                  </jats:inline-formula> is a Riesz potential, <jats:inline-formula>
                     <jats:alternatives>
                        <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_math-2021-0025_eq_003.png" />
                        <m:math xmlns:m="http://www.w3.org/1998/Math/MathML">
                           <m:mn>0</m:mn>
                           <m:mo><</m:mo>
                           <m:mi>α</m:mi>
                           <m:mo><</m:mo>
                           <m:mi>N</m:mi>
                        </m:math>
                        <jats:tex-math>0\lt \alpha \lt N</jats:tex-math>
                     </jats:alternatives>
                  </jats:inline-formula>, and <jats:inline-formula>
                     <jats:alternatives>
                        <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_math-2021-0025_eq_004.png" />
                        <m:math xmlns:m="http://www.w3.org/1998/Math/MathML">
                           <m:mstyle displaystyle="true">
                              <m:mfrac>
                                 <m:mrow>
                                    <m:mi>N</m:mi>
                                    <m:mo>+</m:mo>
                                    <m:mi>α</m:mi>
                                 </m:mrow>
                                 <m:mrow>
                                    <m:mi>N</m:mi>
                                 </m:mrow>
                              </m:mfrac>
                           </m:mstyle>
                           <m:mo><</m:mo>
                           <m:mi>p</m:mi>
                           <m:mo><</m:mo>
                           <m:mstyle displaystyle="true">
                              <m:mfrac>
                                 <m:mrow>
                                    <m:mi>N</m:mi>
                                    <m:mo>+</m:mo>
                                    <m:mi>α</m:mi>
                                 </m:mrow>
                                 <m:mrow>
                                    <m:mi>N</m:mi>
                                    <m:mo>−</m:mo>
                                    <m:mn>2</m:mn>
                                 </m:mrow>
                              </m:mfrac>
                           </m:mstyle>
                        </m:math>
                        <jats:tex-math>\displaystyle \frac{N+\alpha }{N}\lt p\lt \displaystyle \frac{N+\alpha }{N-2}</jats:tex-math>
                     </jats:alternatives>
                  </jats:inline-formula>, with <jats:inline-formula>
                     <jats:alternatives>
                        <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_math-2021-0025_eq_005.png" />
                        <m:math xmlns:m="http://www.w3.org/1998/Math/MathML">
                           <m:mi>λ</m:mi>
                           <m:mo>></m:mo>
                           <m:mn>0</m:mn>
                        </m:math>
                        <jats:tex-math>\lambda \gt 0</jats:tex-math>
                     </jats:alternatives>
                  </jats:inline-formula>. Under suitable assumption on <jats:inline-formula>
                     <jats:alternatives>
                        <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_math-2021-0025_eq_006.png" />
                        <m:math xmlns:m="http://www.w3.org/1998/Math/MathML">
                           <m:mi>V</m:mi>
                        </m:math>
                        <jats:tex-math>V</jats:tex-math>
                     </jats:alternatives>
                  </jats:inline-formula> and <jats:inline-formula>
                     <jats:alternatives>
                        <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_math-2021-0025_eq_007.png" />
                        <m:math xmlns:m="http://www.w3.org/1998/Math/MathML">
                           <m:mi>K</m:mi>
                        </m:math>
                        <jats:tex-math>K</jats:tex-math>
                     </jats:alternatives>
                  </jats:inline-formula>, we research the existence of positive solutions of the equations. Furthermore, we obtain the asymptotic behavior of solutions as <jats:inline-formula>
                     <jats:alternatives>
                        <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_math-2021-0025_eq_008.png" />
                        <m:math xmlns:m="http://www.w3.org/1998/Math/MathML">
                           <m:mi>λ</m:mi>
                           <m:mo>→</m:mo>
                           <m:mn>0</m:mn>
                        </m:math>
                        <jats:tex-math>\lambda \to 0</jats:tex-math>
                     </jats:alternatives>
                  </jats:inline-formula>.</jats:p>

where I α is a Riesz potential, α N 0 < < , and N α N p N α N 2 + < < + − , with λ 0 > . Quasilinear Schrödinger equations of the form which has been called the superfluid film equation in plasma physics by Kurihara in [1] (cf. [2,3]). In the case h s s 1 in the theory of Heisenberg ferromagnets and magnons [9], in dissipative quantum mechanics, and in condensed matter theory [10,11]. For more details, we refer the readers to [12,13] and the references therein.
It is remarkable that there are few papers investigating quasilinear equation with singularity. To the best of our knowledge, it only appears in [32], J. Marcos do Ó and A. Moameni established the singular quasilinear Schrödinger equation where Ω is a ball in N (N 2 ≥ ) centered at the origin, α 0 1 < < . Furthermore, they obtained the existence of radially symmetric positive solutions by taking advantage of Nehari manifold and some techniques about implicit function theorem when λ belongs to a certain neighborhood of the first eigenvalue λ 1 of the eigenvalue problem In [33], they studied the following Choquard-type quasilinear Schrödinger equation: , V : N → is radial potential, and I α is a Riesz potential. They consider the existence of ground state solutions.
To the best of our knowledge, there seems to be little progress on the existence of a positive solution for quasilinear Choquard equation with singularity. By the motivation of the above work, in our study, we establish the existence of a positive solution for problem (1.1) with singularity. First, the nonlinearity of problem (1.1) is nonlocal, and it is much more difficult to obtain the existence of positive solutions. Second, we investigate the relationships between quasilinear Choquard equation involving and without convolution, which makes our studies more interesting. At last, we obtain the asymptotic behavior of solutions as λ 0 → . Before stating our main result, we suppose that the functions V x ( ) and K x ( ) satisfy the following assumptions: x Vx μ meas : Now, we state our main results as follows.
The remainder of this paper is organized as follows. In Section 2, some preliminary results are presented, and in Section 3, we give the proof of our main results.

Variational setting and preliminaries
To prove our conclusion, we give some basic notations and preliminaries. First, we can rewrite (1.1) as It may also be noted that we can not apply directly the variational method to study (1.1), since the natural associated functional I given by is not well defined in general. We make the changing of variables w f u If we make the change of variable u f w = ( ), we may rewrite equation I u ( ) in the form It can be easily proved that the functional w λ ( ) is of class C 1 (see [34]) in E. Moreover, the critical points of λ are weak solutions of the equation be a Hilbert space endowed with inner product and norm We denote by • p ∥ ∥ the usual L p -norm in the sequel for convenience, where p 1 ≤ ≤ +∞. In this step, we see that (1.1) is variational and its weak solutions are the critical points of the functional given by [35]) The function f satisfies the following properties: (A1) f is uniquely defined C ∞ function and invertible; ( To prove Theorem 1.1, we need the following results. Proof. By the Sobolev inequality, Hölder inequality and Lemma 2.1 A 6 ( ) yield For any w E ∈ , using (2.1) and (3.1), for λ 0 > and γ Since γ 0, 1 ∈ ( ), λ is coercive and bounded from below on E for each λ 0 Then, by the weakly lower semi-continuity of the norm, Lemma 2.4 in [36] and (3.4), we obtain (1) We will prove that for any Since by the Beppolevi Monotone convergence Theorem and Lemma 2.1(A 10 ), we have (2) We show that w 0 0 > in N and w 0 is a solution of problem (1.1). Given ε 0 > , define g ε ε R : , λ 0 0 ( ) = ( + ). Then, g attains its minimum at t 0 = by Lemma 3.1, which implies that Taking ε 0 → + to the above inequality and based on the fact that Ω 0 ε → as ε 0 → + , we get The above inequality also holds for v − ; hence, we have Analogous to the proof of [32, Theorem 1], we obtain w C N 0 loc 2 ∈ ( ). Since w 0 0 ≥ , the strong maximum principle implies w 0 0 > , and w E 0 ∈ is a solution of problem (1.1).
(3) We show that the solution w 0 is unique. Assume that w E ∈ is also a solution, then for any φ E ∈ Subtracting (3.9) and (3.10), since K x 0 ( ) > , it follows from Lemma 2.4 (see [36]) and λ 0 > that Passing to the lim inf as n → ∞ in (3.11), by (3.12), (3.13) and the weakly lower semi-continuity of the definition, we have