A refined bound on the dimension of ℝ N for an elliptic system involving critical terms with infinitely many solutions
Abstract: In this paper, we extend the result of Yan and Yang [16] on equations to an elliptic system involving critical Sobolev and Hardy–Sobolev exponents in bounded domains satisfying some geometric condition. In addition, we weaken the conditions on the dimension N and on the potential {a(x)} set in [16]. Our main result asserts, by a variational global-compactness argument, that the condition on the dimension N can be refined from {N\geq 7} to {N>\max(4,\lfloor 2s\rfloor+2)}, where {0<s<2} and still end up… Show more
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Cited by 2 publications
References 14 publications
On a critical elliptic system with concave –convex nonlinearities
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On a critical elliptic system with concave–convex nonlinearities
No abstract
In this article, we study a Hardy-Sobolev critical elliptic system involving coupled perturbation terms: (0.1) − Δ u + V 1 ( x ) u = η 1 η 1 + η 2 ∣ u ∣ η 1 − 2 u ∣ v ∣ η 2 ∣ x ′ ∣ + α α + β Q ( x ) ∣ u ∣ α − 2 u ∣ v ∣ β , − Δ v + V 2 ( x ) v = η 2 η 1 + η 2 ∣ v ∣ η 2 − 2 v ∣ u ∣ η 1 ∣ x ′ ∣ + β α + β Q ( x ) ∣ v ∣ β − 2 v ∣ u ∣ α , \left\{\begin{array}{c}-\Delta u+{V}_{1}\left(x)u=\frac{{\eta }_{1}}{{\eta }_{1}+{\eta }_{2}}\frac{{| u| }^{{\eta }_{1}-2}u{| v| }^{{\eta }_{2}}}{| x^{\prime} | }+\frac{\alpha }{\alpha +\beta }Q\left(x)| u{| }^{\alpha -2}u| v{| }^{\beta },\\ -\Delta v+{V}_{2}\left(x)v=\frac{{\eta }_{2}}{{\eta }_{1}+{\eta }_{2}}\frac{{| v| }^{{\eta }_{2}-2}v{| u| }^{{\eta }_{1}}}{| x^{\prime} | }+\frac{\beta }{\alpha +\beta }Q\left(x){| v| }^{\beta -2}v{| u| }^{\alpha },\end{array}\right. where n ≥ 3 n\ge 3 , 2 ≤ m < n 2\le m\lt n , x ≔ ( x ′ , x ″ ) ∈ R m × R n − m x:= \left(x^{\prime} ,{x}^{^{\prime\prime} })\in {{\mathbb{R}}}^{m}\times {{\mathbb{R}}}^{n-m} , η 1 , η 2 > 1 {\eta }_{1},{\eta }_{2}\gt 1 , and η 1 + η 2 = 2 ( n − 1 ) n − 2 {\eta }_{1}+{\eta }_{2}=\frac{2\left(n-1)}{n-2} , α , β > 1 \alpha ,\beta \gt 1 and α + β < 2 n n − 2 \alpha +\beta \lt \frac{2n}{n-2} , and V 1 ( x ) , V 2 ( x ) , Q ( x ) ∈ C ( R n ) {V}_{1}\left(x),{V}_{2}\left(x),Q\left(x)\in C\left({{\mathbb{R}}}^{n}) . Observing that (0.1) is doubly coupled, we first develop two efficient tools (i.e., a refined Sobolev inequality and a variant of the “Vanishing” lemma). On the previous tools, we will establish a global compactness result (i.e., a complete description for the Palais-Smale sequences of the corresponding energy functional) and some existence result for (0.1) via variational method. Our strategy turns out to be very concise because we avoid the use of Levy concentration functions and truncation techniques.
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