Using the Mountain-Pass Theorem of Ambrosetti and Rabinowitz we prove thatThe technique is based on the existence of extremals of some Hardy-Sobolev type embeddings of independent interest. We also show that if u ∈ D p 1 (R n ) is a weak solution in R n of −∆pu − µ|x| −p |u| p−2 u = |x| −s |u| p ⋆ (s)−2 u + |u| q−2 u, then u ≡ 0 when either 1 < q < p ⋆ , or q > p ⋆ and u is also of class L ∞ loc (R n \ {0}).
In this paper, we give sufficient conditions for the existence and nonexistence of nonnegative nontrivial entire weak solutions of p-Laplacian elliptic inequalities, with possibly singular weights and gradient terms, of the form div {g(|x|)|Du|p-2Du} ≥ h(|x|)f(u)ℓ(|Du|). We achieve our conclusions by using a generalized version of the well-known Keller–Ossermann condition, first introduced in [2] for the generalized mean curvature case, and in [11, Sec. 4] for the nonweighted p-Laplacian equation. Several existence results are also proved in Secs. 2 and 3, from which we deduce simple criteria of independent interest stated in the Introduction.
Existence and non-existence results are established for quasilinear elliptic problems with nonlinear boundary conditions and lack of compactness. The proofs combine variational methods with the geometrical feature, due to the competition between the different growths of the non-linearities.
We consider the elliptic and parabolic superquadratic diffusive Hamilton-Jacobi equations: ∆u + |∇u| p = 0 and ut = ∆u + |∇u| p , with p > 2 and homogeneous Dirichlet conditions. For the elliptic problem in a half-space, we prove a Liouville-type classification, or symmetry result, which asserts that any solution has to be one-dimensional. This turns out to be an efficient tool to study the behavior of boundary gradient blow-up (GBU) solutions of the parabolic problem in general bounded domains of R n with smooth boundaries.Namely, we show that in a neighborhood of the boundary, at leading order, solutions display a global ODE type behavior of the form uνν ∼ −u p ν , with domination of the normal derivatives upon the tangential derivatives. This leads to the existence of a universal, sharp blow-up profile in the normal direction at any GBU point, and moreover implies that the behavior in the tangential direction is more singular. A description of the space-time profile is also obtained. The ODE type behavior and its connection with the Liouvilletype theorem can be considered as an analogue of the well-known results of Merle and Zaag [31] for the subcritical semilinear heat equation, with the significant difference that for the latter, the ODE behavior is in the time direction (instead of the normal spatial direction).On the other hand, it is known that any GBU solution admits a weak continuation, under the form of a global viscosity solution. As another consequence, we show that these viscosity solutions generically lose boundary conditions after GBU. Namely, solutions without loss of boundary conditions after GBU are exceptional and can be characterized as thresholds between global classical solutions and GBU solutions which lose boundary conditions. This result, as well as the above GBU profile, were up to now essentially known only in one space-dimension.Finally, in the case of elliptic Dirichlet problems, we deduce from our Liouville theorem an optimal Bernstein-type estimate, which gives a partial improvement of a local estimate of P.-L. Lions [28].
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