This paper deals with existence and multiplicity of positive solutions to the following class of nonlocal equations with critical nonlinearity: ( − Δ ) s u − γ u | x | 2 s = K ( x ) | u | 2 s ∗ ( t ) − 2 u | x | t + f ( x ) in R N , u ∈ H ˙ s ( R N ) , $$\begin{array}{} \displaystyle \begin{cases} (-{\it\Delta})^s u -\gamma\dfrac{u}{|x|^{2s}}=K(x)\dfrac{|u|^{2^*_s(t)-2}u}{|x|^t}+f(x) \quad\mbox{in}\quad\mathbb R^N,\\ \qquad\qquad\qquad\quad u\in \dot{H}^s(\mathbb R^N), \end{cases} \end{array}$$ where N > 2s, s ∈ (0, 1), 0 ≤ t < 2s < N and 2 s ∗ ( t ) := 2 ( N − t ) N − 2 s $\begin{array}{} \displaystyle 2^*_s(t):=\frac{2(N-t)}{N-2s} \end{array}$ . Here 0 < γ < γ N,s and γ N,s is the best Hardy constant in the fractional Hardy inequality. The coefficient K is a positive continuous function on ℝ N , with K(0) = 1 = lim|x|→∞ K(x). The perturbation f is a nonnegative nontrivial functional in the dual space Ḣs (ℝ N )′ of Ḣs (ℝ N ). We establish the profile decomposition of the Palais-Smale sequence associated with the functional. Further, if K ≥ 1 and ∥f∥(Ḣs )′ is small enough (but f ≢ 0), we establish existence of at least two positive solutions to the above equation.
This paper deals with existence, uniqueness and multiplicity of positive solutions to the following nonlocal system of equations: ( − Δ ) s u = α 2 s * | u | α − 2 u | v | β + f ( x ) in R N , ( − Δ ) s v = β 2 s * | v | β − 2 v | u | α + g ( x ) in R N , u , v > 0 in R N , ( S ) where 0 < s < 1, N > 2s, α, β > 1, α + β = 2N/(N − 2s), and f, g are nonnegative functionals in the dual space of H ˙ s ( R N ) , i.e., 〈 ( H ˙ s ) ′ f , u 〉 H ˙ s ⩾ 0 , whenever u is a nonnegative function in H ˙ s ( R N ) . When f = 0 = g, we show that the ground state solution of ( S ) is unique. On the other hand, when f and g are nontrivial nonnegative functionals with ker(f) = ker(g), then we establish the existence of at least two different positive solutions of ( S ) provided that ‖ f ‖ ( H ˙ s ) ′ and ‖ g ‖ ( H ˙ s ) ′ are small enough. Moreover, we also provide a global compactness result, which gives a complete description of the Palais–Smale sequences of the above system.
We study existence and multiplicity of positive solutions of the following class of nonlocal scalar field equations:where s ∈ (0, 1),We prove existence of a positive solution when f ≡ 0 under certain asymptotic behavior on the function a. Moreover, when a(x) ≥ 1, a(x) → 1 as |x| → ∞ and f H −s (R N ) is small enough (but f ≡ 0), then we show that (P) admits at least two positive solutions. Finally, we establish existence of three positive solutions to (P), under the condition that a(x) ≤ 1 with a(x) → 1 as |x| → ∞ and f H −s (R N ) is small enough (but f ≡ 0).
This paper deals with existence and multiplicity of positive solutions to the following class of nonlocal equations with critical nonlinearity:whereN−2s . Here 0 < γ < γ N,s and γ N,s is the best Hardy constant in the fractional Hardy inequality. The coefficient K is a positive continuous function on R N , with K(0) = 1 = lim |x|→∞ K(x). The perturbation f is a nonnegative nontrivial functional in the dual space Ḣs (R N ) of Ḣs (R N ) i.e., ( Ḣs ) ′ f, u Ḣs ≥ 0, whenever u is a nonnegative function in Ḣs (R N ). We establish the profile decomposition of the Palais-Smale sequence associated with the functional. Further, if K ≥ 1 and f ( Ḣs ) ′ is small enough (but f ≡ 0), we establish existence of at least two positive solutions to the above equation.
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