critical sobolev exponents
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2020 ◽  
Vol 120 (3-4) ◽  
pp. 199-248
Author(s):  
Jianhua Chen ◽  
Xianjiu Huang ◽  
Dongdong Qin ◽  
Bitao Cheng

In this paper, we study the following generalized quasilinear Schrödinger equation − div ( ε 2 g 2 ( u ) ∇ u ) + ε 2 g ( u ) g ′ ( u ) | ∇ u | 2 + V ( x ) u = K ( x ) | u | p − 2 u + | u | 22 ∗ − 2 u , x ∈ R N , where N ⩾ 3, ε > 0, 4 < p < 22 ∗ , g ∈ C 1 ( R , R + ), V ∈ C ( R N ) ∩ L ∞ ( R N ) has a positive global minimum, and K ∈ C ( R N ) ∩ L ∞ ( R N ) has a positive global maximum. By using a change of variable, we obtain the existence and concentration behavior of ground state solutions for this problem with critical growth, and establish a phenomenon of exponential decay. Moreover, by Ljusternik–Schnirelmann theory, we also prove the existence of multiple solutions.


Author(s):  
Pasquale Candito ◽  
Salvatore A. Marano ◽  
Kanishka Perera

We obtain a pair of nontrivial solutions for a class of concave-linear-convex type elliptic problems that are either critical or subcritical. The solutions we find are neither local minimizers nor of mountain pass type in general. They are higher critical points in the sense that they each have a higher critical group that is nontrivial. This fact is crucial for showing that our solutions are nontrivial. We also prove some intermediate results of independent interest on the localization and homotopy invariance of critical groups of functionals involving critical Sobolev exponents.


2018 ◽  
Vol 61 (3) ◽  
pp. 705-733 ◽  
Author(s):  
Athanasios N. Lyberopoulos

AbstractWe are concerned with the existence of positive weak solutions, as well as the existence of bound states (i.e. solutions inW1,p(ℝN)), for quasilinear scalar field equations of the form$$ - \Delta _pu + V(x) \vert u \vert ^{p - 2}u = K(x) \vert u \vert ^{q - 2}u + \vert u \vert ^{p^ * - 2}u,\qquad x \in {\open R}^N,$$where Δpu: =div(|∇u|p−2∇u), 1 <p<N,p*: =Np/(N−p) is the critical Sobolev exponent,q∈ (p, p*), whileV(·) andK(·) are non-negative continuous potentials that may decay to zero as |x| → ∞ but are free from any integrability or symmetry assumptions.


2018 ◽  
Vol 2018 ◽  
pp. 1-10
Author(s):  
Jing Li ◽  
Caisheng Chen

We focus on the following elliptic system with critical Sobolev exponents:  -div⁡∇up-2∇u+m(x)up-2u=λup⁎-2u+(1/η)Gu(u,v),  x∈RN; -div⁡∇vq-2∇v+n(x)vq-2v=μvq⁎-2v+(1/η)Gv(u,v),  x∈RN; u(x)>0,v(x)>0,  x∈RN, where μ,λ>0,1<p≤q<N, either η∈(1,p) or η∈(q,p⁎), and critical Sobolev exponents p⁎=pN/(N-p) and q⁎=qN/(N-q). Conditions on potential functions m(x),n(x) lead to no compact embedding. Relying on concentration-compactness principle, mountain pass lemma, and genus theory, the existence of solutions to the elliptic system with η∈(q,p⁎) or η∈(1,p) will be established.


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