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Classification of positive solutions to a system of Hardy-Sobolev type equations

Research paper by WeiDAI, ZhaoLIU

Indexed on: 28 Oct '17Published on: 01 Sep '17Published in: Acta Mathematica Scientia



Abstract

In this paper, we are concerned with the following Hardy-Sobolev type system (0.1){(-Δ)α2u(x)=υq(x)|y|t2(-Δ)α2υ(x)=up(x)|y|t1,x=(y,z)∈(ℝk\{0})×ℝn-k,<math class="math"><mrow is="true"><mrow is="true"><mo is="true">{</mo><mrow is="true"><mtable is="true"><mtr is="true"><mtd is="true"><mrow is="true"><msup is="true"><mrow is="true"><mrow is="true"><mo is="true">(</mo><mrow is="true"><mo is="true">-</mo><mi is="true">Δ</mi></mrow><mo is="true">)</mo></mrow></mrow><mrow is="true"><mfrac is="true"><mi is="true">α</mi><mn is="true">2</mn></mfrac></mrow></msup><mi is="true">u</mi><mrow is="true"><mo is="true">(</mo><mi is="true">x</mi><mo is="true">)</mo></mrow><mo is="true">=</mo><mfrac is="true"><mrow is="true"><msup is="true"><mi is="true">υ</mi><mi is="true">q</mi></msup><mrow is="true"><mo is="true">(</mo><mi is="true">x</mi><mo is="true">)</mo></mrow></mrow><mrow is="true"><msup is="true"><mrow is="true"><mrow is="true"><mo is="true">|</mo><mi is="true">y</mi><mo is="true">|</mo></mrow></mrow><mrow is="true"><msub is="true"><mi is="true">t</mi><mn is="true">2</mn></msub></mrow></msup></mrow></mfrac></mrow></mtd></mtr><mtr is="true"><mtd is="true"><mrow is="true"><msup is="true"><mrow is="true"><mrow is="true"><mo is="true">(</mo><mrow is="true"><mo is="true">-</mo><mi is="true">Δ</mi></mrow><mo is="true">)</mo></mrow></mrow><mrow is="true"><mfrac is="true"><mi is="true">α</mi><mn is="true">2</mn></mfrac></mrow></msup><mi is="true">υ</mi><mrow is="true"><mo is="true">(</mo><mi is="true">x</mi><mo is="true">)</mo></mrow><mo is="true">=</mo><mfrac is="true"><mrow is="true"><msup is="true"><mi is="true">u</mi><mi is="true">p</mi></msup><mrow is="true"><mo is="true">(</mo><mi is="true">x</mi><mo is="true">)</mo></mrow></mrow><mrow is="true"><msup is="true"><mrow is="true"><mrow is="true"><mo is="true">|</mo><mi is="true">y</mi><mo is="true">|</mo></mrow></mrow><mrow is="true"><msub is="true"><mi is="true">t</mi><mn is="true">1</mn></msub></mrow></msup></mrow></mfrac><mo is="true">,</mo></mrow></mtd></mtr></mtable></mrow></mrow><mi is="true">x</mi><mo is="true">=</mo><mrow is="true"><mo is="true">(</mo><mrow is="true"><mi is="true">y</mi><mo is="true">,</mo><mi is="true">z</mi></mrow><mo is="true">)</mo></mrow><mo is="true">∈</mo><mrow is="true"><mo is="true">(</mo><mrow is="true"><msup is="true"><mtext is="true">ℝ</mtext><mi is="true">k</mi></msup><mo is="true">\</mo><mrow is="true"><mo is="true">{</mo><mn is="true">0</mn><mo is="true">}</mo></mrow></mrow><mo is="true">)</mo></mrow><mo is="true">×</mo><msup is="true"><mtext is="true">ℝ</mtext><mrow is="true"><mi is="true">n</mi><mo is="true">-</mo><mi is="true">k</mi></mrow></msup><mo is="true">,</mo></mrow></math>where 0<α<n, 0<t1,t2 < min{α,k}, and 1<p≤τ1:=n+α-2t1n-α,1<q≤τ2:=n+α-2t2n-α.<math class="math"><mrow is="true"><mn is="true">1</mn><mo is="true">&lt;</mo><mi is="true">p</mi><mo is="true">≤</mo><msub is="true"><mi is="true">τ</mi><mn is="true">1</mn></msub><mo is="true">:</mo><mo is="true">=</mo><mfrac is="true"><mrow is="true"><mi is="true">n</mi><mo is="true">+</mo><mi is="true">α</mi><mo is="true">-</mo><mn is="true">2</mn><msub is="true"><mi is="true">t</mi><mn is="true">1</mn></msub></mrow><mrow is="true"><mi is="true">n</mi><mo is="true">-</mo><mi is="true">α</mi></mrow></mfrac><mo is="true">,</mo><mn is="true">1</mn><mo is="true">&lt;</mo><mi is="true">q</mi><mo is="true">≤</mo><msub is="true"><mi is="true">τ</mi><mn is="true">2</mn></msub><mo is="true">:</mo><mo is="true">=</mo><mfrac is="true"><mrow is="true"><mi is="true">n</mi><mo is="true">+</mo><mi is="true">α</mi><mo is="true">-</mo><mn is="true">2</mn><msub is="true"><mi is="true">t</mi><mn is="true">2</mn></msub></mrow><mrow is="true"><mi is="true">n</mi><mo is="true">-</mo><mi is="true">α</mi></mrow></mfrac><mo is="true">.</mo></mrow></math>. We first establish the equivalence of classical and weak solutions between PDE system (0.2){u(x)=∫ℝnGα(x,ξ)up(ξ)|η|t2dξυ(x)=∫ℝnGα(x,ξ)up(ξ)|η|t2dξ,<math class="math"><mrow is="true"><mrow is="true"><mo is="true">{</mo><mrow is="true"><mtable is="true"><mtr is="true"><mtd is="true"><mrow is="true"><mi is="true">u</mi><mrow is="true"><mo is="true">(</mo><mi is="true">x</mi><mo is="true">)</mo></mrow><mo is="true">=</mo><msub is="true"><mo is="true">∫</mo><mrow is="true"><msup is="true"><mtext is="true">ℝ</mtext><mi is="true">n</mi></msup></mrow></msub><mrow is="true"><msub is="true"><mi is="true">G</mi><mi is="true">α</mi></msub><mrow is="true"><mo is="true">(</mo><mrow is="true"><mi is="true">x</mi><mo is="true">,</mo><mi is="true">ξ</mi></mrow><mo is="true">)</mo></mrow><mfrac is="true"><mrow is="true"><msup is="true"><mi is="true">u</mi><mi is="true">p</mi></msup><mrow is="true"><mo is="true">(</mo><mi is="true">ξ</mi><mo is="true">)</mo></mrow></mrow><mrow is="true"><msup is="true"><mrow is="true"><mrow is="true"><mo is="true">|</mo><mi is="true">η</mi><mo is="true">|</mo></mrow></mrow><mrow is="true"><msub is="true"><mi is="true">t</mi><mn is="true">2</mn></msub></mrow></msup></mrow></mfrac><mtext is="true">d</mtext><mi is="true">ξ</mi></mrow></mrow></mtd></mtr><mtr is="true"><mtd is="true"><mrow is="true"><mi is="true">υ</mi><mrow is="true"><mo is="true">(</mo><mi is="true">x</mi><mo is="true">)</mo></mrow><mo is="true">=</mo><msub is="true"><mo is="true">∫</mo><mrow is="true"><msup is="true"><mtext is="true">ℝ</mtext><mi is="true">n</mi></msup></mrow></msub><mrow is="true"><msub is="true"><mi is="true">G</mi><mi is="true">α</mi></msub><mrow is="true"><mo is="true">(</mo><mrow is="true"><mi is="true">x</mi><mo is="true">,</mo><mi is="true">ξ</mi></mrow><mo is="true">)</mo></mrow><mfrac is="true"><mrow is="true"><msup is="true"><mi is="true">u</mi><mi is="true">p</mi></msup><mrow is="true"><mo is="true">(</mo><mi is="true">ξ</mi><mo is="true">)</mo></mrow></mrow><mrow is="true"><msup is="true"><mrow is="true"><mrow is="true"><mo is="true">|</mo><mi is="true">η</mi><mo is="true">|</mo></mrow></mrow><mrow is="true"><msub is="true"><mi is="true">t</mi><mn is="true">2</mn></msub></mrow></msup></mrow></mfrac><mtext is="true">d</mtext><mi is="true">ξ</mi></mrow><mo is="true">,</mo></mrow></mtd></mtr></mtable></mrow></mrow></mrow></math>where Gα(x,ξ)=cn,α|x-ξ|n-α<math class="math"><mrow is="true"><msub is="true"><mi is="true">G</mi><mi is="true">α</mi></msub><mrow is="true"><mo is="true">(</mo><mrow is="true"><mi is="true">x</mi><mo is="true">,</mo><mi is="true">ξ</mi></mrow><mo is="true">)</mo></mrow><mo is="true">=</mo><mfrac is="true"><mrow is="true"><msub is="true"><mi is="true">c</mi><mrow is="true"><mi is="true">n</mi><mo is="true">,</mo><mi is="true">α</mi></mrow></msub></mrow><mrow is="true"><msup is="true"><mrow is="true"><mrow is="true"><mo is="true">|</mo><mrow is="true"><mi is="true">x</mi><mo is="true">-</mo><mi is="true">ξ</mi></mrow><mo is="true">|</mo></mrow></mrow><mrow is="true"><mi is="true">n</mi><mo is="true">-</mo><mi is="true">α</mi></mrow></msup></mrow></mfrac></mrow></math> is the Green's function of (-Δ)α2<math class="math"><mrow is="true"><msup is="true"><mrow is="true"><mrow is="true"><mo is="true">(</mo><mrow is="true"><mo is="true">-</mo><mi is="true">Δ</mi></mrow><mo is="true">)</mo></mrow></mrow><mrow is="true"><mfrac is="true"><mi is="true">α</mi><mn is="true">2</mn></mfrac></mrow></msup></mrow></math> in ℝn. Then, by the method of moving planes in the integral forms, in the critical case p = τ1 and q = τ2, we prove that each pair of nonnegative solutions(u,v) of (0.1) is radially symmetric and monotone decreasing about the origin in ℝk and some point z0 in ℝn-k. In the subcritical case n-t1p+1+n-t2q+1>n-α,1<p≤τ1<math class="math"><mrow is="true"><mfrac is="true"><mrow is="true"><mi is="true">n</mi><mo is="true">-</mo><msub is="true"><mi is="true">t</mi><mn is="true">1</mn></msub></mrow><mrow is="true"><mi is="true">p</mi><mo is="true">+</mo><mn is="true">1</mn></mrow></mfrac><mo is="true">+</mo><mfrac is="true"><mrow is="true"><mi is="true">n</mi><mo is="true">-</mo><msub is="true"><mi is="true">t</mi><mn is="true">2</mn></msub></mrow><mrow is="true"><mi is="true">q</mi><mo is="true">+</mo><mn is="true">1</mn></mrow></mfrac><mo is="true">&gt;</mo><mi is="true">n</mi><mo is="true">-</mo><mi is="true">α</mi><mo is="true">,</mo><mn is="true">1</mn><mo is="true">&lt;</mo><mi is="true">p</mi><mo is="true">≤</mo><msub is="true"><mi is="true">τ</mi><mn is="true">1</mn></msub></mrow></math> and 1 < q ≤ τ2, we derive the nonexistence of nontrivial nonnegative solutions for (0.1).