Existence and nonexistence of nontrivial solutions for critical biharmonic equations

Existence and concentration of solutions for a class of biharmonic Kirchhoff equations with discontinuous nonlinearity

“…For problem () with the term $$$ b=0 $$$ and $$$ {V}_0=V(x) $$$, Sun et al 15 investigated a class of nonlinear biharmonic problem with $$$ p $$$‐Laplacian under some hypotheses on $$$ V(x) $$$ and $$$ f\left(x,v\right) $$$, and they derive the existence and multiplicity of nontrivial solutions by variational methods and Gagliardo–Nirenberg inequality. He and Lv 16 considered problem () with $$$ V(x)=\lambda, b=0 $$$ and $$$ f(u)={\left|u\right|}^{2^{\ast \ast }-2}u $$$; they proved the existence and nonexistence of nontrivial solutions to problem (). In their study, 17 Song and Shi dealt with problem () in bounded domains; the existence and multiplicity of solutions are derived by employing the concentration‐compactness principle and variational method.…”

Existence of nontrivial positive solutions for generalized quasilinear elliptic equations with critical exponent

“…7), we haveo(1) = J ′ ψ (u n ), u n − u = Ω |∆u n | 2 dx − Ω ∆u n ∆udx + ψ 0 ( Ω |∇u n | 2 dx) Ω ∇u n ∇(u n − u)dx − λ Ω f (x, u n )(u n − u)dx − Ω |u n | 2 * * −2 u n (u n − u)dx.…”