For an even functional on a Banach space, the symmetric mountain pass lemma gives a sequence of critical values which converges to zero. Under the same assumptions on the functional, this paper establishes a new critical point theorem which provides a sequence of critical points converging to zero. The theorem is applied to sublinear elliptic equations. Then a sequence of solutions converging to zero is obtained.

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The asymptotic behavior of viscosity solutions to the Cauchy-Dirichlet problem for the degenerate parabolic equation u t = ∆ ∞ u in Ω × (0, ∞), where ∆ ∞ stands for the so-called infinity-Laplacian, is studied in three cases: (i) Ω = R N and the initial data has a compact support; (ii) Ω is bounded and the boundary condition is zero; (iii) Ω is bounded and the boundary condition is non-zero. Our method of proof is based on the comparison principle and barrier function arguments. Explicit representations of separable type and self-similar type of solutions are also established. Moreover, in case (iii), we propose another type of barrier function deeply related to a solution of ∆ ∞ φ = 0.

Every solution u = u(x, t) of the Cauchy-Dirichlet problem for the fast diffusion equation, ∂ t (|u| m−2 u) = ∆u in Ω × (0, ∞) with a smooth bounded domain Ω of R N and 2 < m < 2 * := 2N/(N − 2) + , vanishes in finite time at a power rate. This paper is concerned with asymptotic profiles of sign-changing solutions and a stability analysis of the profiles. Our method of proof relies on a detailed analysis of a dynamical system on some surface in the usual energy space as well as energy method and variational method.

In this paper, we study the (p, q)-Laplace equation in a bounded domain under the Dirichlet boundary condition. We give a sufficient condition of the nonlinear term for the existence of a sequence of solutions converging to zero or diverging to infinity. Moreover, we give a priori estimates of the C 1-norms of solutions under a suitable condition on the nonlinear term.