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In this article, we investigate the existence of multiple positive solutions to the following multi-critical Schrödinger equation: (0.1) − Δ u + λ V ( x ) u = μ ∣ u ∣ p − 2 u + ∑ i = 1 k ( ∣ x ∣ − ( N − α i ) ∗ ∣ u ∣ 2 i ∗ ) ∣ u ∣ 2 i ∗ − 2 u in R N , u ∈ H 1 ( R N ) , \left\{\begin{array}{l}-\Delta u+\lambda V\left(x)u=\mu | u{| }^{p-2}u+\mathop{\displaystyle \sum }\limits_{i=1}^{k}\left(| x{| }^{-\left(N-{\alpha }_{i})}\ast | u{| }^{{2}_{i}^{\ast }})| u{| }^{{2}_{i}^{\ast }-2}u\hspace{1.0em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{N},\hspace{1.0em}\\ u\hspace{0.33em}\in {H}^{1}\left({{\mathbb{R}}}^{N}),\hspace{1.0em}\end{array}\right. where λ , μ ∈ R + , N ≥ 4 \lambda ,\mu \in {{\mathbb{R}}}^{+},N\ge 4 , and 2 i ∗ = N + α i N − 2 {2}_{i}^{\ast }=\frac{N+{\alpha }_{i}}{N-2} with N − 4 < α i < N N-4\lt {\alpha }_{i}\lt N , i = 1 , 2 , … , k i=1,2,\ldots ,k are critical exponents and 2 < p < 2 min ∗ = min { 2 i ∗ : i = 1 , 2 , … , k } 2\lt p\lt {2}_{\min }^{\ast }={\rm{\min }}\left\{{2}_{i}^{\ast }:i=1,2,\ldots ,k\right\} . Suppose that Ω = int V − 1 ( 0 ) ⊂ R N \Omega ={\rm{int}}\hspace{0.33em}{V}^{-1}\left(0)\subset {{\mathbb{R}}}^{N} is a bounded domain, we show that for λ \lambda large, problem (0.1) possesses at least cat Ω ( Ω ) {{\rm{cat}}}_{\Omega }\left(\Omega ) positive solutions.

In this paper, we show the existence of nontrivial solutions for doubly critical nonlocal elliptic problems in ℝ N {\mathbb{R}^{N}} involving fractional Laplacians. Our approach is elementary without the use of neither the concentration-compactness principle nor the extension argument developed in [4].

In this paper, we show that the minimizing problem Λ s , N , k , α = inf u ∈ H ˙ s ⁢ ( ℝ N ) , u ≢ 0 ⁡ ∫ ℝ N | ( - Δ ) s 2 ⁢ u ⁢ ( x ) | 2 ⁢ 𝑑 x ( ∫ ℝ N | u ⁢ ( x ) | 2 s , α * | y | α ⁢ 𝑑 x ) 2 2 s , α * $\Lambda_{s,N,k,\alpha}=\inf_{u\in\dot{H}^{s}(\mathbb{R}^{N}),u\not\equiv 0}% \frac{\int_{\mathbb{R}^{N}}|(-\Delta)^{\frac{s}{2}}u(x)|^{2}\,dx}{\bigl{(}\int% _{\mathbb{R}^{N}}\frac{|u(x)|^{2^{*}_{s,\alpha}}}{|y|^{\alpha}}\,dx\bigr{)}^{% \frac{2}{2^{*}_{s,\alpha}}}}$ is achieved by a positive, cylindrically symmetric and strictly decreasing function u ⁢ ( x ) ${u(x)}$ provided 0 < s < N 2 ${0 , 0 < α < 2 ⁢ s ${0 , where x = ( y , z ) ∈ ℝ k × ℝ N - k ${x=(y,z)\in\mathbb{R}^{k}\times\mathbb{R}^{N-k}}$ and 2 s , α * = 2 ⁢ ( N - α ) N - 2 ⁢ s ${2^{*}_{s,\alpha}=\frac{2(N-\alpha)}{N-2s}}$ . Decaying laws for the minimizer u are also established.

This paper is concerned with the following H’enon type problem where B1(0) is the unit ball in ℝN centered at the origin, 1 < p, q and if N ≥ 3. We show that there exists α∗ > 0 such that the ground state solution of the problem is non-radial if α > α∗. We also consider the limiting behavior of the ground state solution (u, v) as p + q → 2∗. We prove that the maximum points of two components u and v concentrate at the same point on the boundary ∂B1(0).

We consider embeddings of a class of fractional Sobolev spaces Es(Ω) (s ≥ 0) into their limiting Lp(Ω) spaces, for a bounded smooth domain Ω ⊂ ℝN, and study the relationship between the best constants for these embeddings and other best constants corresponding to the case Ω = ℝN.

We study the existence of solutions for the semilinear elliptic equation where 1 < p < , Q(x) is a sign-changing function such that both the supports of Q+ and Q- may have infinite measure. We show that the problem has at least one solution.

We consider the question of existence of nonnegative solutions of a class of p-Laplacian elliptic problems in the whole space ℝN having a sign-changing nonlinearity of the form a(x)g(u).