Existence and multiplicity of solutions for fractional p-Laplacian equation involving critical concave-convex nonlinearities

We investigate the following fractional p -Laplacian convex-concave problem: <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block" overflow="scroll"> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:msub> <m:mrow> <m:mi>P</m:mi> </m:mrow> <m:mrow> <m:mi>λ</m:mi> </m:mrow> </m:msub> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mspace class="nbsp" width="0.3333em"/> <m:mspace class="nbsp" width="0.3333em"/> <m:mspace class="nbsp" width="0.3333em"/> <m:mspace class="nbsp" width="0.3333em"/> <m:mspace class="nbsp" width="0.3333em"/> <m:mspace class="nbsp" width="0.3333em"/> <m:mspace class="nbsp" width="0.3333em"/> <m:mspace class="nbsp" width="0.3333em"/> <m:mspace class="nbsp" width="0.3333em"/> <m:mspace class="nbsp" width="0.3333em"/> <m:mfenced close="" open="{"> <m:mrow> <m:mtable class="cases"> <m:mtr> <m:mtd columnalign="left"> <m:mtable class="aligned"> <m:mtr> <m:mtd columnalign="right"> <m:msubsup> <m:mrow> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mo>−</m:mo> <m:mi mathvariant="normal">Δ</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mrow> <m:mi>p</m:mi> </m:mrow> <m:mrow> <m:mi>s</m:mi> </m:mrow> </m:msubsup> <m:mi>u</m:mi> </m:mtd> <m:mtd columnalign="left"> <m:mo>=</m:mo> <m:mi>λ</m:mi> <m:mo stretchy="false">|</m:mo> <m:mi>u</m:mi> <m:msup> <m:mrow> <m:mo stretchy="false">|</m:mo> </m:mrow> <m:mrow> <m:mi>q</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:mo stretchy="false">|</m:mo> <m:mi>u</m:mi> <m:msup> <m:mrow> <m:mo stretchy="false">|</m:mo> </m:mrow> <m:mrow> <m:msubsup> <m:mrow> <m:mi>p</m:mi> </m:mrow> <m:mrow> <m:mi>s</m:mi> </m:mrow> <m:mrow> <m:mo>*</m:mo> </m:mrow> </m:msubsup> <m:mo>−</m:mo> <m:mn>2</m:mn> </m:mrow> </m:msup> <m:mi>u</m:mi> </m:mtd> <m:mtd columnalign="right"/> <m:mtd columnalign="left"> <m:mtext> </m:mtext> <m:mtext>in</m:mtext> <m:mspace class="nbsp" width="0.3333em"/> <m:mi mathvariant="normal">Ω</m:mi> <m:mo>,</m:mo> </m:mtd> </m:mtr> <m:mtr> <m:mtd columnalign="right"> <m:mi>u</m:mi> </m:mtd> <m:mtd columnalign="left"> <m:mo>=</m:mo> <m:mn>0</m:mn> <m:mspace class="nbsp" width="0.3333em"/> <m:mspace class="nbsp" width="0.3333em"/> <m:mspace class="nbsp" width="0.3333em"/> <m:mspace class="nbsp" width="0.3333em"/> <m:mspace class="nbsp" width="0.3333em"/> <m:mspace class="nbsp" width="0.3333em"/> <m:mspace class="nbsp" width="0.3333em"/> <m:mspace class="nbsp" width="0.3333em"/> <m:mspace class="nbsp" width="0.3333em"/> <m:mspace class="nbsp" width="0.3333em"/> <m:mspace class="nbsp" width="0.3333em"/> <m:mspace class="nbsp" width="0.3333em"/> <m:mspace class="nbsp" width="0.3333em"/> <m:mspace class="nbsp" width="0.3333em"/> <m:mspace class="nbsp" width="0.3333em"/> <m:mspace class="nbsp" width="0.3333em"/> <m:mspace class="nbsp" width="0.3333em"/> <m:mspace class="nbsp" width="0.3333em"/> <m:mspace class="nbsp" width="0.3333em"/> <m:mspace class="nbsp" width="0.3333em"/> <m:mspace class="nbsp" width="0.3333em"/> </m:mtd> <m:mtd columnalign="right"/> <m:mtd columnalign="left"> <m:mtext> </m:mtext> <m:mtext>in</m:mtext> <m:mspace class="nbsp" width="0.3333em"/> <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:mi mathvariant="normal">Ω</m:mi> <m:mo>,</m:mo> </m:mtd> </m:mtr> </m:mtable> <m:mtext> </m:mtext> </m:mtd> </m:mtr> </m:mtable> </m:mrow> </m:mfenced> </m:math> $$({P}λ) {cases}{aligned} {(-{Δ})}ₚˢu& =λ u{ }q-2u+ u{ }^{{p}ₛ^{{}}-2}u & & in {Ω}, \\ u& =0 & & in {R}ⁿ{}{Ω}, {aligned} {cases}$$ where Ω is a bounded C 1,1 domain in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <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:math> Rⁿ , s ∈ (0, 1), p > q > 1, n > sp , λ > 0, and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msubsup> <m:mrow> <m:mi>p</m:mi> </m:mrow> <m:mrow> <m:mi>s</m:mi> </m:mrow> <m:mrow> <m:mo>*</m:mo> </m:mrow> </m:msubsup> <m:mo>=</m:mo> <m:mfrac> <m:mrow> <m:mi>n</m:mi> <m:mi>p</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> <m:mo>−</m:mo> <m:mi>s</m:mi> <m:mi>p</m:mi> </m:mrow> </m:mfrac> </m:math> pₛ^=np/n-sp is the critical Sobolev exponent. Our analysis extends classical works (A. Ambrosetti, H. Brezis, and G. Cerami, “Combined effects of concave and convex nonlinearities in some elliptic problems,” J. Funct. Anal. , vol. 122, no. 2, pp. 519–543, 1994, B. Barrios, E. Colorado, R. Servadei, and F. Soria, “A critical fractional equation with concave-convex power nonlinearities,” Ann. Inst. Henri Poincare Anal. Non Lineaire , vol. 32, no. 4, pp. 875–900, 2015, J. García Azorero, J. Manfredi, and I. Peral Alonso, “Sobolev versus Hölder local minimizer and global multiplicity for some quasilinear elliptic equations,” Commun. Contemp. Math. , vol. 2, no. 3, pp. 385–404, 2000) to fractional p -Laplacian. Owing to the nonlinear and nonlocal properties of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msubsup> <m:mrow> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mo>−</m:mo> <m:mi mathvariant="normal">Δ</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mrow> <m:mi>p</m:mi> </m:mrow> <m:mrow> <m:mi>s</m:mi> </m:mrow> </m:msubsup> </m:math> (-Δ)ₚˢ , we need to overcome many difficulties and apply notably different approaches, due to the lack of Picone identity, the stability theory, and the strong comparison principle. We show first a dichotomy result: a positive <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msubsup> <m:mrow> <m:mi>W</m:mi> </m:mrow> <m:mrow> <m:mn>0</m:mn> </m:mrow> <m:mrow> <m:mi>s</m:mi> <m:mo>,</m:mo> <m:mi>p</m:mi> </m:mrow> </m:msubsup> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi mathvariant="normal">Ω</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> W₀s,p(Ω) solution of ( P λ ) exists if and only if λ ∈ (0, Λ] with an extremal value Λ ∈ (0, ∞). The <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msubsup> <m:mrow> <m:mi>W</m:mi> </m:mrow> <m:mrow> <m:mn>0</m:mn> </m:mrow> <m:mrow> <m:mi>s</m:mi> <m:mo>,</m:mo> <m:mi>p</m:mi> </m:mrow> </m:msubsup> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi mathvariant="normal">Ω</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> W₀s,p(Ω) regularity for the extremal solution seems to be unknown regardless of whether s = 1 or s ∈ (0, 1). When p ≥ 2, p − 1 < q < p and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi>n</m:mi> <m:mo>></m:mo> <m:mfrac> <m:mrow> <m:mi>s</m:mi> <m:mi>p</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>q</m:mi> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mrow> <m:mi>q</m:mi> <m:mo>+</m:mo> <m:mn>1</m:mn> <m:mo>−</m:mo> <m:mi>p</m:mi> </m:mrow> </m:mfrac> </m:math> n >sp(q+1)/q+1-p , we get two positive solutions for ( P λ ) with small λ > 0. Here the mountain pass structure is more involved than the classical situations due to the lack of explicit minimizers for the Sobolev embedding, we should proceed carefully and simultaneously the construction of mountain pass geometry and the estimate for mountain pass level. Finally, we show another new result for ( P λ ) and all p > q > 1: without sign constraint, ( P λ ) possesses infinitely many solutions when λ > 0 is small enough. Here we use the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msub> <m:mrow> <m:mi mathvariant="double-struck">Z</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> </m:math> Z₂ -genus theory, based on a space decomposition for reflexible and separable Banach spaces, which has its own interest.