Existence and multiplicity of solutions for a nonlocal problem with critical Sobolev exponent
Abstract In this work, we are interested in considering the following nonlocal problem − a + b ∫ Ω | ∇ u | 2 d x Δ u = μ | u | 2 ∗ − 2 u + λ | u | q − 2 u , x ∈ Ω , u = 0 , x ∈ ∂ Ω , where Ω ⊂ R N ( N ≥ 4 ) is a smooth bounded domain, a ≥ 0 , b > 0 , 1 q 2 , μ , λ > 0 and 2 ∗ = 2 N N − 2 is the critical Sobolev exponent. By using the variational method and the critical point theorem, some existence and multiplicity results are obtained.
Multiplicity and concentration of positive solutions for a Kirchhoff type problem with critical growth
Abstract In this paper we concern with the multiplicity and concentration of positive solutions for the semilinear Kirchhoff type equation { − ( e 2 a + b e ∫ R 3 | ∇ u | 2 ) Δ u + M ( x ) u = λ f ( u ) + | u | 4 u , x ∈ R 3 , u ∈ H 1 ( R 3 ) , u > 0 , x ∈ R 3 , where e > 0 is a small parameter, a, b are positive constants and λ > 0 is a parameter, and f is a continuous superlinear and subcritical nonlinearity. Suppose that M ( x ) has at least one minimum. We first prove that the system has a positive ground state solution u e for λ > 0 sufficiently large and e > 0 sufficiently small. Then we show that u e converges to the positive ground state solution of the associated limit problem and concentrates to a minimum point of M ( x ) in certain sense as e → 0 . Moreover, some further properties of the ground state solutions are also studied. Finally, we investigate the relation between the number of positive solutions and the topology of the set of the global minima of the potentials by minimax theorems and the Ljusternik–Schnirelmann theory.
General expression for the density effect for the ionization loss of charged particles
The equations for the density-effect correction $\ensuremath{\delta}(X)$ for the ionization loss as given in a previous paper have been extended to the case where $\ensuremath{\delta}({X}_{0})\ensuremath{\equiv}{\ensuremath{\delta}}_{0}g0$. Here ${X}_{0}$ is the lower limit of the range ${X}_{0}lXl{X}_{1}$, where a particularly simple expression for $\ensuremath{\delta}(X)$ applies; $X={log}_{10}(\frac{p}{{m}_{0}c})$, where $p$ is the momentum and ${m}_{0}$ is the mass of the incident particle. The case ${\ensuremath{\delta}}_{0}g0$ applies directly to the calculation of $\ensuremath{\delta}(X)$ for conductors (metals).