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On singular equations with critical and supercritical exponents

Research paper by MousomiBhaktaa, SanjibanSantrab

Indexed on: 11 Nov '17Published on: 01 Sep '17Published in: Journal of Differential Equations



Abstract

We study the problem(Iε){−Δu−μu|x|2=up−εuqin Ω,u>0in Ω,u∈H01(Ω)∩Lq+1(Ω),<math class="math"><mo stretchy="false" is="true">(</mo><msub is="true"><mrow is="true"><mi is="true">I</mi></mrow><mrow is="true"><mi is="true">ε</mi></mrow></msub><mo stretchy="false" is="true">)</mo><mrow is="true"><mo stretchy="true" is="true">{</mo><mtable displaystyle="true" columnspacing="0.2em" is="true"><mtr is="true"><mtd columnalign="right" is="true"><mo is="true">−</mo><mi mathvariant="normal" is="true">Δ</mi><mi is="true">u</mi><mo is="true">−</mo><mfrac is="true"><mrow is="true"><mi is="true">μ</mi><mi is="true">u</mi></mrow><mrow is="true"><mo stretchy="false" is="true">|</mo><mi is="true">x</mi><msup is="true"><mrow is="true"><mo stretchy="false" is="true">|</mo></mrow><mrow is="true"><mn is="true">2</mn></mrow></msup></mrow></mfrac></mtd><mtd columnalign="left" is="true"><mo is="true">=</mo><msup is="true"><mrow is="true"><mi is="true">u</mi></mrow><mrow is="true"><mi is="true">p</mi></mrow></msup><mo is="true">−</mo><mi is="true">ε</mi><msup is="true"><mrow is="true"><mi is="true">u</mi></mrow><mrow is="true"><mi is="true">q</mi></mrow></msup><mspace width="1em" is="true"></mspace><mtext is="true">in&nbsp;</mtext><mspace width="1em" is="true"></mspace><mi mathvariant="normal" is="true">Ω</mi><mo is="true">,</mo></mtd></mtr><mtr is="true"><mtd columnalign="right" is="true"><mi is="true">u</mi></mtd><mtd columnalign="left" is="true"><mo is="true">&gt;</mo><mn is="true">0</mn><mspace width="1em" is="true"></mspace><mtext is="true">in&nbsp;</mtext><mspace width="1em" is="true"></mspace><mi mathvariant="normal" is="true">Ω</mi><mo is="true">,</mo></mtd></mtr><mtr is="true"><mtd columnalign="right" is="true"><mi is="true">u</mi></mtd><mtd columnalign="left" is="true"><mo is="true">∈</mo><msubsup is="true"><mrow is="true"><mi is="true">H</mi></mrow><mrow is="true"><mn is="true">0</mn></mrow><mrow is="true"><mn is="true">1</mn></mrow></msubsup><mo stretchy="false" is="true">(</mo><mi mathvariant="normal" is="true">Ω</mi><mo stretchy="false" is="true">)</mo><mo is="true">∩</mo><msup is="true"><mrow is="true"><mi is="true">L</mi></mrow><mrow is="true"><mi is="true">q</mi><mo is="true">+</mo><mn is="true">1</mn></mrow></msup><mo stretchy="false" is="true">(</mo><mi mathvariant="normal" is="true">Ω</mi><mo stretchy="false" is="true">)</mo><mo is="true">,</mo></mtd></mtr></mtable></mrow></math> where ⁎q>p≥2⁎−1<math class="math"><mi is="true">q</mi><mo is="true">&gt;</mo><mi is="true">p</mi><mo is="true">≥</mo><msup is="true"><mrow is="true"><mn is="true">2</mn></mrow><mrow is="true"><mo is="true">⁎</mo></mrow></msup><mo is="true">−</mo><mn is="true">1</mn></math>, ε>0<math class="math"><mi is="true">ε</mi><mo is="true">&gt;</mo><mn is="true">0</mn></math>, Ω⊆RN<math class="math"><mi mathvariant="normal" is="true">Ω</mi><mo is="true">⊆</mo><msup is="true"><mrow is="true"><mi mathvariant="double-struck" is="true">R</mi></mrow><mrow is="true"><mi is="true">N</mi></mrow></msup></math> is a bounded domain with smooth boundary, 0∈Ω<math class="math"><mn is="true">0</mn><mo is="true">∈</mo><mi mathvariant="normal" is="true">Ω</mi></math>, N≥3<math class="math"><mi is="true">N</mi><mo is="true">≥</mo><mn is="true">3</mn></math> and 0<μ<μ¯:=(N−22)2<math class="math"><mn is="true">0</mn><mo is="true">&lt;</mo><mi is="true">μ</mi><mo is="true">&lt;</mo><mover accent="true" is="true"><mrow is="true"><mi is="true">μ</mi></mrow><mrow is="true"><mo stretchy="false" is="true">¯</mo></mrow></mover><mo is="true">:</mo><mo is="true">=</mo><msup is="true"><mrow is="true"><mo stretchy="true" maxsize="2.4ex" minsize="2.4ex" is="true">(</mo><mfrac is="true"><mrow is="true"><mi is="true">N</mi><mo is="true">−</mo><mn is="true">2</mn></mrow><mn is="true">2</mn></mfrac><mo stretchy="true" maxsize="2.4ex" minsize="2.4ex" is="true">)</mo></mrow><mrow is="true"><mn is="true">2</mn></mrow></msup></math>. We completely classify the singularity of solution at 0 in the supercritical case. Using the transformation v=|x|νu<math class="math"><mi is="true">v</mi><mo is="true">=</mo><mo stretchy="false" is="true">|</mo><mi is="true">x</mi><msup is="true"><mrow is="true"><mo stretchy="false" is="true">|</mo></mrow><mrow is="true"><mi is="true">ν</mi></mrow></msup><mi is="true">u</mi></math>, we reduce the problem (Iε)<math class="math"><mo stretchy="false" is="true">(</mo><msub is="true"><mrow is="true"><mi is="true">I</mi></mrow><mrow is="true"><mi is="true">ε</mi></mrow></msub><mo stretchy="false" is="true">)</mo></math> to (Jε)<math class="math"><mo stretchy="false" is="true">(</mo><msub is="true"><mrow is="true"><mi is="true">J</mi></mrow><mrow is="true"><mi is="true">ε</mi></mrow></msub><mo stretchy="false" is="true">)</mo></math>(Jε){−div(|x|−2ν∇v)=|x|−(p+1)νvp−ε|x|−(q+1)νvqin Ω,v>0in Ω,v∈H01(Ω,|x|−2ν)∩Lq+1(Ω,|x|−(q+1)ν),<math class="math"><mo stretchy="false" is="true">(</mo><msub is="true"><mrow is="true"><mi is="true">J</mi></mrow><mrow is="true"><mi is="true">ε</mi></mrow></msub><mo stretchy="false" is="true">)</mo><mrow is="true"><mo stretchy="true" is="true">{</mo><mtable displaystyle="true" columnspacing="0.2em" is="true"><mtr is="true"><mtd columnalign="right" is="true"><mo is="true">−</mo><mi is="true">d</mi><mi is="true">i</mi><mi is="true">v</mi><mo stretchy="false" is="true">(</mo><mo stretchy="false" is="true">|</mo><mi is="true">x</mi><msup is="true"><mrow is="true"><mo stretchy="false" is="true">|</mo></mrow><mrow is="true"><mo is="true">−</mo><mn is="true">2</mn><mi is="true">ν</mi></mrow></msup><mi mathvariant="normal" is="true">∇</mi><mi is="true">v</mi><mo stretchy="false" is="true">)</mo></mtd><mtd columnalign="left" is="true"><mo is="true">=</mo><mo stretchy="false" is="true">|</mo><mi is="true">x</mi><msup is="true"><mrow is="true"><mo stretchy="false" is="true">|</mo></mrow><mrow is="true"><mo is="true">−</mo><mo stretchy="false" is="true">(</mo><mi is="true">p</mi><mo is="true">+</mo><mn is="true">1</mn><mo stretchy="false" is="true">)</mo><mi is="true">ν</mi></mrow></msup><msup is="true"><mrow is="true"><mi is="true">v</mi></mrow><mrow is="true"><mi is="true">p</mi></mrow></msup><mo is="true">−</mo><mi is="true">ε</mi><mo stretchy="false" is="true">|</mo><mi is="true">x</mi><msup is="true"><mrow is="true"><mo stretchy="false" is="true">|</mo></mrow><mrow is="true"><mo is="true">−</mo><mo stretchy="false" is="true">(</mo><mi is="true">q</mi><mo is="true">+</mo><mn is="true">1</mn><mo stretchy="false" is="true">)</mo><mi is="true">ν</mi></mrow></msup><msup is="true"><mrow is="true"><mi is="true">v</mi></mrow><mrow is="true"><mi is="true">q</mi></mrow></msup><mspace width="1em" is="true"></mspace><mtext is="true">in&nbsp;</mtext><mspace width="1em" is="true"></mspace><mi mathvariant="normal" is="true">Ω</mi><mo is="true">,</mo></mtd></mtr><mtr is="true"><mtd columnalign="right" is="true"><mi is="true">v</mi></mtd><mtd columnalign="left" is="true"><mo is="true">&gt;</mo><mn is="true">0</mn><mspace width="1em" is="true"></mspace><mtext is="true">in&nbsp;</mtext><mspace width="1em" is="true"></mspace><mi mathvariant="normal" is="true">Ω</mi><mo is="true">,</mo></mtd></mtr><mtr is="true"><mtd columnalign="right" is="true"><mi is="true">v</mi></mtd><mtd columnalign="left" is="true"><mo is="true">∈</mo><msubsup is="true"><mrow is="true"><mi is="true">H</mi></mrow><mrow is="true"><mn is="true">0</mn></mrow><mrow is="true"><mn is="true">1</mn></mrow></msubsup><mo stretchy="false" is="true">(</mo><mi mathvariant="normal" is="true">Ω</mi><mo is="true">,</mo><mo stretchy="false" is="true">|</mo><mi is="true">x</mi><msup is="true"><mrow is="true"><mo stretchy="false" is="true">|</mo></mrow><mrow is="true"><mo is="true">−</mo><mn is="true">2</mn><mi is="true">ν</mi></mrow></msup><mo stretchy="false" is="true">)</mo><mo is="true">∩</mo><msup is="true"><mrow is="true"><mi is="true">L</mi></mrow><mrow is="true"><mi is="true">q</mi><mo is="true">+</mo><mn is="true">1</mn></mrow></msup><mo stretchy="false" is="true">(</mo><mi mathvariant="normal" is="true">Ω</mi><mo is="true">,</mo><mo stretchy="false" is="true">|</mo><mi is="true">x</mi><msup is="true"><mrow is="true"><mo stretchy="false" is="true">|</mo></mrow><mrow is="true"><mo is="true">−</mo><mo stretchy="false" is="true">(</mo><mi is="true">q</mi><mo is="true">+</mo><mn is="true">1</mn><mo stretchy="false" is="true">)</mo><mi is="true">ν</mi></mrow></msup><mo stretchy="false" is="true">)</mo><mo is="true">,</mo></mtd></mtr></mtable></mrow></math> and then formulating a variational problem for (Jε)<math class="math"><mo stretchy="false" is="true">(</mo><msub is="true"><mrow is="true"><mi is="true">J</mi></mrow><mrow is="true"><mi is="true">ε</mi></mrow></msub><mo stretchy="false" is="true">)</mo></math>, we establish the existence of a variational solution vε<math class="math"><msub is="true"><mrow is="true"><mi is="true">v</mi></mrow><mrow is="true"><mi is="true">ε</mi></mrow></msub></math> and characterize the asymptotic behavior of vε<math class="math"><msub is="true"><mrow is="true"><mi is="true">v</mi></mrow><mrow is="true"><mi is="true">ε</mi></mrow></msub></math> as ε→0<math class="math"><mi is="true">ε</mi><mo stretchy="false" is="true">→</mo><mn is="true">0</mn></math> by variational arguments when ⁎p=2⁎−1<math class="math"><mi is="true">p</mi><mo is="true">=</mo><msup is="true"><mrow is="true"><mn is="true">2</mn></mrow><mrow is="true"><mo is="true">⁎</mo></mrow></msup><mo is="true">−</mo><mn is="true">1</mn></math>.