On domains with conical points, weighted Sobolev spaces with powers of the distance to the conical points as weights form a classical framework for describing the regularity of solutions of elliptic boundary value problems, cf. papers by Kondrat’ev and Maz’ya-Plamenevskii. Two classes of weighted norms are usually considered: Homogeneous norms, where the weight exponent varies with the order of...

In this work, by using variational methods we study the existence of nontrivial positive solutions for a class p-Kirchhoff type problems with critical Sobolev exponent.

where ua = 1 vol(Mn) ∫ Mn u, p∗ = np/(n − p) is the Sobolev conjugate of p. This inequality can be proved by combining Sobolev inequality with Poincaré inequality, see, for example, Hebey’s book [8]. In this paper we are interested in the estimates of the best constant and the existence of extremal functions to the above inequality. Analytically these are naturally motivated questions. On the o...

In this paper, we consider the existence and asymptotic properties of solutions to following Kirchhoff equation \(- \left(a+b\int_{{\mathbb{R}^3}} {{{\left| {\nabla u} \right|}^2}}\right) \Delta u=\lambda u+ {| u |^{p - 2}}u+\mu |^{q 2}}u\) in \(\mathbb{R}^{3}\) under normalized constraint \(\int_{{\mathbb{R}^3}} {{u}^2}=c^2\), where \(a>0\), \(b>0\), \(c>0\), \(2<q<\frac{14}{3}&...

In this paper, we consider the following nonlinear Schrödinger equations with mixed nonlinearities: { ? ? u = ? + ? | q 2 ? in R N , ? H 1 ( ) ? a where ? 3 > 0 and < . We prove paper (1) Existence of solutions mountain-pass type for 4 ; (2) nonexistence ground states ? large; (3) Precisely asymptotic behaviors as ? goes to its upper bound. Our studies answer some open questions proposed by Soa...