Quasilinear problems with critical Sobolev exponent for the Grushin p-Laplace operator

Entire nodal solutions to the pure critical exponent problem for the p-Laplacian

The main purpose of this paper is to establish the existence and multiplicity of nontrivial solutions for a quasilinear Neumann problem with critical exponent. It is shown, by the methods of the Lions concentration-compactness principle and the mountain pass lemma, that under certain conditions, the existence of nontrivial solutions are obtained.

In this paper we study a quasilinear elliptic problem which involves the critical Sobolev exponent and multiple Hardy-type terms. By the analytic technics and variational methods, the existence and nonexistence of the ground states for the problem is established under certain assumptions.

The main theorem establishes the existence of a positive decaying solution u ∈ D 0 1,p (Rn) of a quasilinear elliptic problem involving the p-Laplacian operator and the critical Sobolev exponent pN/(N - p), 1<p<N. The conclusion depends on the existence of a lowest eigenvalue of a related quasilinear eigenvalue problem. A preliminary result yields a Palais-Smale compactness condition for an associated functional via concentration-compactness methods of P. L. Lions.

We investigate minimum energy paths of the quasi-linear problem with the p-Laplacian operator and a double-well potential. We adapt the String method of E, Ren, and Vanden-Eijnden (J. Chem. Phys. 126, 2007) to locate saddle-type solutions. In one-dimension, the String method is shown to find a minimum energy path that can align along one-dimensional ridges of saddle-continua. We then apply the same method to locate saddle solutions and transition paths of the two-dimensional quasi-linear problem. The method developed is applicable to a general class of quasi-linear PDEs.

We establish the multiplicity of positive solutions to a quasilinear Neumann problem in expanding balls and hemispheres with critical exponent in the boundary condition.

In this paper, we are concerned with a kind of quasilinear elliptic problems, which involves the Caffarelli-Kohn-Nirenberg inequality and critical exponents. By employing varia- tional methods and analytical techniques, the existence of sign-changing solutions to the problem is proved.

A compactness result is revised in order to prove the pointwise convergence of the gradients of a sequence of solutions to a general quasilinear inequality (anisotropic or not, degenerate or not) and for an arbitrary open set. Combining this result with the well-known Brézis-Lieb lemma, we derive simple proofs of Palais-Smale properties in many optimization problems especially on unbounded domains.

Existence and concentration of ground states to a quasilinear problem with competing potentials

Multiple solutions for the p&q-Laplacian problem with critical exponent

We study the following elliptic system with critical exponent: Here, Ω is a smooth bounded domain of ℝ N (N ≥ 6), is the critical Sobolev exponent, 0 < λ1, λ2 < λ1(Ω) and μ1, μ2 > 0, where λ1(Ω) is the first eigenvalue of − Δ with the Dirichlet boundary condition. When β = 0, this turn to be the well-known Brézis-Nirenberg problem. We show that, for each fixed β <0, this system has a sign-changing solution in the following sense: one component changes sign and has exactly two nodal domains, while the other one is positive. We also study the asymptotic behavior of these solutions as β → − ∞ and phase separation appears. Precisely, two components of these solutions tend to repel each other and converge to solutions of the Brézis-Nirenberg problem in segregated regions.

We investigate the solvability of the Neumann problem involving two critical exponents: Sobolev and Hardy-Sobolev. We establish the existence of a solution in three cases: (i) 2 < p + 1 < 2*(s), (ii) p + 1 = 2*(s) and (iii) 2*(s) < p + 1 ≤ 2*, where 2*(s) = 2(N-s)/N-2, 0 < s < 2, and 2* = 2(N-s)/N-2 denote the critical Hardy-Sobolev exponent and the critical Sobolev exponent, respectively.

On an elliptic problem with critical exponent and Hardy potential

This paper concerns itself with the nonexistence and multiplicity of solutions for the following fractional Kirchhoff-type problem involving the critical Sobolev exponent: [ a + b ⁢ ( ∬ ℝ 2 ⁢ N | u ⁢ ( x ) - u ⁢ ( y ) | p | x - y | N + p ⁢ s ⁢ 𝑑 x ⁢ 𝑑 y ) θ - 1 ] ⁢ ( - Δ ) p s ⁢ u = | u | p s * - 2 ⁢ u + λ ⁢ f ⁢ ( x ) in ⁢ ℝ N , \Biggl{[}a+b\biggl{(}\iint_{\mathbb{R}^{2N}}\frac{\lvert u(x)-u(y)\rvert^{p}}{% \lvert x-y\rvert^{N+ps}}\,dx\,dy\biggr{)}^{\theta-1}\Biggr{]}(-\Delta)_{p}^{s}% u=\lvert u\rvert^{p_{s}^{*}-2}u+\lambda f(x)\quad\text{in }\mathbb{R}^{N}, where a ≥ 0 {a\kern-1.0pt\geq\kern-1.0pt0} , b &gt; 0 , θ &gt; 1 {b\kern-1.0pt&gt;\kern-1.0pt0,\theta\kern-1.0pt&gt;\kern-1.0pt1} , ( - Δ ) p s {(-\Delta)_{p}^{s}} is the fractional p-Laplacian with 0 &lt; s &lt; 1 {0\kern-1.0pt&lt;\kern-1.0pts\kern-1.0pt&lt;\kern-1.0pt1} and 1 &lt; p &lt; N / s {1\kern-1.0pt&lt;\kern-1.0ptp\kern-1.0pt&lt;\kern-1.0ptN/s} , p s * = N ⁢ p / ( N - p ⁢ s ) {p_{s}^{*}\kern-1.0pt=\kern-1.0ptNp/(N-ps)} is the critical Sobolev exponent, λ ≥ 0 {\lambda\geq 0} is a parameter, and f ∈ L p s * / ( p s * - 1 ) ⁢ ( ℝ N ) ∖ { 0 } {f\in L^{p_{s}^{*}/(p_{s}^{*}-1)}(\mathbb{R}^{N})\setminus\{0\}} is a nonnegative function. When λ = 0 {\lambda=0} , we show that the multiplicity and nonexistence of solutions for the above problem are related with N, θ, s, p, a, and b. When λ &gt; 0 {\lambda&gt;0} , by using Ekeland’s variational principle and the mountain pass theorem, we show that there exists λ * * &gt; 0 {\lambda^{**}&gt;0} such that the above problem admits at least two nonnegative solutions for all λ ∈ ( 0 , λ * * ) {\lambda\in(0,\lambda^{**})} . In the latter case, in order to overcome the loss of compactness, we derive a fractional version of the principle of concentration compactness in the setting of the fractional p-Laplacian.

In this paper, we study the positive radial solutions for the Hénon equations with weighted critical exponents on the unit ball in with . We first confirm that with is the critical exponent for the embedding from into and name as the Hénon‐Sobolev critical exponent. Then, following the great ideas of Brezis and Nirenberg (Comm Pure Appl Math. 1983;36:437‐477), we establish the existence and nonexistence of positive radial solutions of the problems with single Hénon‐Sobolev critical exponent and linear or nonlinear but subcritical perturbations. We further study the problems with multiple critical exponents, which may be Hénon‐Sobolev critical exponents, Hardy‐Sobolev critical exponents, or Sobolev critical exponents. The methods and arguments involved with are the mountain pass theorem and the strong maximum principle and the Pohozaev identity.