In this paper we consider the quasilinear critical problem(Pλ){−Δpu=λuq−1+up⁎−1inΩu>0inΩu=0on∂Ω where Ω is a regular bounded domain in RN, N≥p2, 1<p<2, p≤q<p⁎, p⁎=Np/(N−p), λ>0 is a parameter. In ...spite of the lack of C2 regularity of the energy functional associated to (Pλ), we employ new Morse techniques to derive a multiplicity result of solutions. We show that there exists λ⁎>0 such that, for each λ∈(0,λ⁎), either (Pλ) has P1(Ω) distinct solutions or there exists a sequence of quasilinear problems approximating (Pλ), each of them having at least P1(Ω) distinct solutions. These results complete those obtained in 23 for the case p≥2.
We study asymptotic behavior of positive ground state solutions of the nonlinear Kirchhoff equation −(a+b∫RN|∇u|2)Δu+λu=uq−1+up−1inRN,(Pλ)as λ→0 and λ→+∞, where N=3 or N=4, 2<q≤p≤2∗, 2∗=2NN−2 is the ...Sobolev critical exponent, a>0, b≥0 are constants and λ>0 is a parameter. In particular, we prove that in the case 2<q<p=2∗, as λ→0, after a suitable rescaling the ground state solutions of (Pλ) converge to the unique positive solution of the equation −Δu+u=uq−1 and as λ→+∞, after another rescaling the ground state solutions of (Pλ) converge to a particular solution of the critical Emden–Fowler equation −Δu=u2∗−1. We establish a sharp asymptotic characterization of such rescalings, which depends in a non-trivial way on the space dimension N=3 and N=4. We also discuss a connection of our results with a mass constrained problem associated to (Pλ) with normalization constraint ∫RN|u|2=c2. As a consequence of the main results, we obtain the existence, non-existence and asymptotic behavior of positive normalized solutions of such a problem. In particular, we obtain the exact number and their precise asymptotic expressions of normalized solutions if c>0 is sufficiently large or sufficiently small. Our results also show that in the space dimension N=3, there is a striking difference between the cases b=0 and b≠0. More precisely, if b≠0, then both p0≔10/3 and pb≔14/3 play a role in the existence, non-existence, the exact number and asymptotic behavior of the normalized solutions of the mass constrained problem, which is completely different from those for the corresponding nonlinear Schrödinger equation and which reveals the special influence of the nonlocal term.
In this paper we study the existence and asymptotic behavior of solutions of $$-\Delta u=\mu\frac{u}{|x|^{2}}+|x|^{\alpha}u^{p(\alpha)-1-\varepsilon},\qquad u>0 \ \text{in}\ B_{R}(0)$$ with Dirichlet ...boundary condition. Here, $-2<\alpha<0$, $p(\alpha)=\frac{2(N+\alpha)}{N-2}$, $0<\varepsilon<p(\alpha)-1$ and $p(\alpha)-1-\varepsilon$ is a nearly critical exponent. We combine variational arguments with the moving plane method to prove the existence of a positive radial solution. Moreover, the asymptotic behaviour of the solutions, as $\varepsilon\to0$, is studied by using ODE techniques.
This paper deals with the following class of nonlocal Schrödinger equations(-\Delta)^s u + u = |u|^{p-1}u in \mathbb{R}^N, for s\in (0,1).We prove existence and symmetry results for the ...solutions $u$ in the fractional Sobolev space H^s(\mathbb{R}^N). Our results are in clear accordance with those for the classical local counterpart, that is when s=1.
In this paper we are concerned with the well-known Brezis-Nirenberg problem <TD NOWRAP ALIGN="CENTER">\begin{displaymath}\begin {cases}-\Delta u= u^{\frac {N+2}{N-2}}+\varepsilon u, ... ...n}~\Omega ...},\\ u=0, &{\text {on}~\partial \Omega }. \end{cases}\end{displaymath} <TD NOWRAP CLASS="eqno" WIDTH="10" ALIGN="RIGHT"> The existence of multi-peak solutions to the above problem for small \varepsilon >0 was obtained (see Monica Musso and Angela Pistoia Indiana Univ. Math. J. 51 (2002), pp. 541-579). However, the uniqueness or the exact number of positive solutions to the above problem is still unknown. Here we focus on the local uniqueness of multi-peak solutions and the exact number of positive solutions to the above problem for small \varepsilon >0. By using various local Pohozaev identities and blow-up analysis, we first detect the relationship between the profile of the blow-up solutions and Green's function of the domain \Omega and then obtain a type of local uniqueness results of blow-up solutions. Lastly we give a description of the number of positive solutions for small positive \varepsilon , which depends also on Green's function.
We study the existence of sign changing solutions to the following problem(0.1){Δu+|u|p−1u=0inΩε;u=0on∂Ωε, where p=n+2n−2 is the critical Sobolev exponent and Ωε is a bounded smooth domain in Rn, ...n≥3, of the form Ωε=Ω\B(0,ε). Here Ω is a smooth bounded domain containing the origin 0 and B(0,ε) denotes the ball centered at the origin with radius ε>0. We construct a new type of sign-changing solutions with high energy to problem (0.1), when the parameter ε is small enough.
In this paper, we study the existence and multiplicity of positive solutions for a Kirchhoff-type problem involving several potentials and critical nonlinearities on unbounded domain. The first main ...result is that the problem possesses a ground state solution concentrating around a concrete set relates to the potentials. The second one is that with the help of Nehari manifold and Ljusternik–Schnirelmann category, the relationship between the number of positive solutions and the category of the global minima set of a suitable ground energy function is established. Moreover, we present a sufficient condition for the nonexistence of ground state solution.
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