On periodic and compactly supported least energy solutions to semilinear elliptic equations with non-Lipschitz nonlinearity

Abstract

We discuss the existence and non-existence of periodic in one variable and compactly supported in the other variables least energy solutions for equations with non-Lipschitz nonlinearity of the form: $- Δ u = λ u^{p} - u^{q}$ in $R^{N + 1}$ , where $0 < q < p < 1$ and $λ \in R$ . The approach is based on the Nehari manifold method supplemented by a one-sided constraint given through the functional of the suitable Pohozaev identity. The limit value of the parameter λ, where the approach is applicable, corresponds to the existence of periodic in one variable and compactly supported in the other variables least energy solutions. This value is found through the extrem values of nonlinear generalized Rayleigh quotients and the so-called curve of the critical exponents of p, q. Important properties of the solutions are derived for suitable ranges of the parameters, such as that they are not trivial with respect to the periodic variable and do not coincide with compactly supported solutions on the entire space $R^{N + 1}$ .

Keywords

Semilinear elliptic equation non-Lipschitz nonlinearity compactly supported solutions periodic solutions generalized Rayleigh’s quotients the Pohozaev identity

1. Introduction

We deal with the following equation: $\begin{aligned} (1.1) & - u_{z z} - Δ_{x} u = λ | u |^{p - 1} u - | u |^{q - 1} u = : f_{λ} (u), (z, x) \in D^{T} : = (- T, T) \times Ω, \end{aligned}$ subject to the periodic boundary conditions in one variable: $\begin{aligned} (1.2) & u (- T, x) = u (T, x), x \in Ω, \\ (1.3) & u_{z} (- T, x) = u_{z} (T, x), x \in Ω, \end{aligned}$ and zero Dirichlet boundary condition on $\partial Ω$ : $\begin{matrix} (1.4) & u (z, x) |_{\partial Ω} = 0, z \in (- T, T) . \end{matrix}$ Here $0 < q < p < 1$ , $0 < T < + \infty$ , $Ω \subset R^{N}$ , $N ⩾ 3$ is a bounded domain with smooth boundary $\partial Ω$ and λ is a real parameter. In addition, we assume that Ω is strictly star-shaped with respect to a point $x_{0} \in R^{N}$ (which will be identified as the origin of coordinates if no confusion arises). We use the notations $Δ_{x} = \partial^{2} / \partial x_{1}^{2} + \dots + \partial^{2} / \partial x_{N}^{2}$ , $u_{z} = \partial u / \partial z$ and $\nabla_{x} : = (\partial / \partial x_{1}, \dots, \partial / \partial x_{N})$ . By weak solution of (1.1)–(1.4) we mean a critical point of the energy functional $\begin{array}{r} Φ_{λ, T} (u) = \frac{1}{2} \int_{D^{T}} (| \nabla_{x} u |^{2} + | u_{z} |^{2}) d x d z - \frac{λ}{p + 1} \int_{D^{T}} | u |^{p + 1} d x d z + \frac{1}{q + 1} \int_{D^{T}} | u |^{q + 1} d x d z, \end{array}$ for $u \in W_{0, per}^{1, 2} (D^{T})$ , where $W_{0, per}^{1, 2} (D^{T})$ is the Sobolev space of functions subject to the periodicity condition (1.2) and zero boundary conditions (1.4), and $D^{T} : = (- T, T) \times Ω$ . The meaning of the periodicity condition (1.3) for a weak solution $u \in W_{0, per}^{1, 2} (D^{T})$ of (1.1) is given below in Section 2.

The construction of solutions that are periodic in part of variables are relevant in a range of applications, including the study of waves on the surface of a deep fluid, the study of convection patterns and gravity–capillary waves in hydrodynamics, in the analysis of self-localized structures without symmetry in non-linear media, vortex flows of an incompressible perfect fluid, and particle-shaped states in field models of elementary particles (see e.g., [1,18,40–42]). Problems of this type often are used as approximation models (in particular, in numerical simulations) in the study of solutions considered on the entire space $R^{N + 1}$ (see e.g., [4,16,33,36,37]).

This subject is also related to the problem of finding new types of solutions, including entire solutions on the whole space which are non-radially symmetric, nonnegative, and do not tend to 0 at infinity. In recent decades, a number of remarkable results for elliptic problems on the existence of various new types of nonnegative solutions including non-radially symmetric in the entire space, compactly supported and partially free boundary solutions have been obtained (see e.g., [5,6,9,10,15,19,23,25,27,28,31,35,38,39]). Furthermore, it has been discovered that finding periodic solutions in part of variables can be useful for the construction of the so-called multiple ends entire and spike-layers solutions [10,31] and has a relation with the construction of Delaunay’s unduloids [4,11,31,32].

We are interested in obtaining a periodically nontrivial solution u of (1.1)–(1.4), i.e., which satisfies $\int_{D^{T}} | u_{z} |^{2} d x d z \neq 0$ . In the case $\int_{D^{T}} | u_{z} |^{2} d x d z = 0$ , we say that u is periodically trivial. By a least energy solution (sometimes also referred to as ground state (cf. [3])) of (1.1)–(1.4) we mean a weak solution u of (1.1)–(1.4) which satisfies the inequality $Φ_{λ, T} (u) ⩽ Φ_{λ, T} (w)$ for any non-zero weak solution $w \in W_{0, per}^{1, 2} (D^{T})$ of (1.1)–(1.3).

The primary aim of the present paper is to analyse the impact of the parameters $λ > 0$ and $T > 0$ on the existence to (1.1)–(1.4) of periodically nontrivial least energy solutions.

To construct solutions periodic in parts of the variables, the approach proposed by Dancer [9] is often used. This approach is based on the fact that solution $ψ^{N}$ of the equation of type (1.1) in the lower dimension, say in $R^{N}$ , can be trivially extended to the whole $R^{N + 1}$ by setting $u_{ext}^{N + 1} (z, x) = ψ^{N} (x)$ , which is in fact, in our terminology periodically trivial. In [9], Dancer using the Crandal-Rabinowitz theorem showed that the solutions, periodic in $z \in (- T, T)$ with some $T > 0$ (as the bifurcation parameter), bifurcate from $u_{ext}^{N + 1} (z, x)$ . Applying this approach requires $f_{λ}$ to be differentiable and $f_{λ}^{'} (0) < 0$ (cf. [9]). However, $f_{λ}$ in (1.1) is non-Lipschitz at zero. An additional obstacle is that the bifurcation methods, as in [9], do not allow to make a conclusion, in a simple way, that the bifurcation branches of solutions obtained consist of least energy solutions.

In the present work, to construct solutions periodic in parts of the variables for (1.1)–(1.4) we use the variational approach proposed in [23–25]. This method makes it possible to find a new type of solutions for elliptic equations in general forms but do not require that $f \in C^{1}$ . Moreover, it can also provide a bifurcation type result [25] and the investigation of the asymptotic behaviour of solutions as T tends to infinity [24].

Since the non-linear term $f_{λ} (u)$ is locally non-Lipschitz, peculiar behavior of solutions of the problem appears. In particular, it may lead to the violation of the Hopf maximum principle on the boundary of Ω and one can expect the existence of a solution periodic on z and compactly supported in Ω, that is, a weak solution $u \in C^{1} (\overline{D^{T}}) \cap C^{2} (D^{T})$ of (1.1)–(1.4) satisfying the supplemented boundary condition $\begin{matrix} (1.5) & \frac{\partial u (z)}{\partial ν} = 0 on \partial Ω, z \in (- T, T) . \end{matrix}$ Here ν denotes the unit outward normal to $\partial Ω$ . See [12,35,38,39] for further details on validity of Hopf maximum principle. It makes sense to refer to such solutions of (1.1)–(1.4) by a more general term as solutions with compact support in part of variables. In what follows, if u is a solution such that property (1.5) is not satisfied we shall call it as a “usual” solution of (1.1)–(1.4).

Thus, one can state the following problem, which is of particular interest to us: Can elliptic equations with non-Lipschitz nonlinearity have solutions with compact support in part of variables?

The existence of compactly supported solutions on all of variables for the equation $\begin{matrix} (1_{R^{M}}^{*}) & - Δ ψ = λ ψ^{p} - ψ^{q} in R^{M}, M ⩾ 1 \end{matrix}$ had been obtained in celebrated results by Kaper and Kwong [27], Kaper, Kwong and Li [28], Cortázar, M. Elgueta and P. Felmer [7,8], J. Serrin and H. Zou [38]. From these results it follows

Theorem 1.1.
Assume $0 < q < p < 1$ , $M ⩾ 2$ and $λ = 1$ . Let $ψ^{M}$ be a nonnegative $C^{1}$ distribution solution of ( 1 R M ∗ ) with connected support. Then the support of $ψ^{M}$ is a ball and $supp (ψ^{M})$ is radially symmetric about the center. Furthermore, equation ( 1 R M ∗ ) admits at most one radial symmetric compactly supported solution and it is a classical, i.e., $ψ^{M} \in C^{2} (R^{M})$ .
Remark 1.1.
Similar result holds for ( 1 R M ∗ ) with any $λ > 0$ since scaling $ψ_{λ}^{M} (y) : = {(σ_{λ})}^{\frac{2}{1 - q}} \cdot ψ^{M} (y / σ_{λ})$ with $σ_{λ} = {(λ)}^{- \frac{1 - q}{2 (p - q)}}$ yields a unique radial symmetric compactly supported solution of ( 1 R M ∗ ) with given λ.

This kind of results can be obtained by using the shooting methods (see [7,27,28]) and the Alexandrov & Serrin moving plane methods (see [8,38]). However, these approaches are difficult to apply directly to the problem (1.1)–(1.4). Indeed, the presence of the separate variable z turns the phase space $H = R$ in the shooting method into an infinite-dimensional space of functions $H = {w (z), z \in (- T, T)}$ , and forces the application of the moving plane method by part of the variables $x \in R^{N}$ . Furthermore, the approaches based on shooting and moving plane methods [7,8,27,28,38] make it difficult to obtain a least energy solution of ( 1 R M ∗ ). We also refer to [2,30] for the existence result of compact support solutions to singular problems by means of suitable sub and super-solution method.

In the present paper, we study the existence of compactly supported solutions by using the variational method introduced in [13–15,21] which allows us to find compactly supported least energy solutions of ( 1 R M ∗ ) using the so-called Pohozaev functional $\begin{array}{rcl} P_{λ, T} (u) & : = & \frac{1}{2_{N}^{}} \int_{D^{T}} | \nabla_{x} u |^{2} d x d z + \frac{1}{2} \int_{D^{T}} | u_{z} |^{2} d x d z + \frac{1}{q + 1} \int_{D^{T}} | u |^{q + 1} d x d z \\ - \frac{λ}{p + 1} \int_{D^{T}} | u |^{p + 1} d x d z, \end{array}$ in introducing a supplementary one-side constraint $P_{λ, T} (u) ⩽ 0$ in the Nehari manifold. Here $2_{N}^{} = 2 N / (N - 2)$ , $N ⩾ 3$ . Thus, we seek the solutions of (1.1)–(1.4) through the minimization of $Φ_{λ, T} (u)$ restricted on the following Nehari manifold subset $\begin{aligned} (1.6) & M_{λ, T} : = {u \in W_{0, per}^{1, 2} (D^{T}) ∖ {0} : Φ_{λ, T}^{'} (u) : = \frac{d}{d t} Φ_{λ, T} (t u) |_{t = 1} = 0, P_{λ, T} (u) ⩽ 0} . \end{aligned}$ Below we show that the solution of (1.1)–(1.4) obtained by this approach yields a least energy solution. It is known [20] that the relevancy of the Nehari manifold method is defined by the so-called extreme values of the applicability of the Nehari manifold method, namely, by the limit points of the set of parameters λ, T, p, q of the problem, where the so-called applicability conditions of the Nehari manifold method $\begin{array}{r} Φ_{λ, T}^{″} (u) : = \frac{d^{2}}{d t^{2}} Φ_{λ, T} (t u) |_{t = 1} \neq 0 and P_{λ, T} (u) < 0, \forall u \in N_{λ} \end{array}$ are satisfied [15,20]. In the present work, we show that the extreme values of the Nehari manifold method for parameters p, q are defined by the so-called curves of critical exponents [21,26], which allow us to introduce the following subset of exponents $\begin{matrix} E_{s} (N) : = {(q, p) \in (0, 1) \times (0, 1) : N (1 - q) (1 - p) - 2 (1 + q) (1 + p) > 0}, \end{matrix}$ delimited by the curve of the critical exponents ${(q, p) \in E_{s} (N) : N (1 - q) (1 - p) - 2 (1 + q) (1 + p) = 0}$ . The main property of $E_{s} (N)$ is that for star-shaped domains Ω in $R^{N}$ , if $(q, p) \in E_{s} (N)$ , any Nehari manifold minimizer $\hat{u} \in M_{λ, T}$ of $Φ_{λ, T}$ is non degenerate, i.e., satisfies $Φ_{λ, T}^{″} (\hat{u}) > 0$ .

To find the extreme values of the applicability of the Nehari manifold method for the parameter λ, we apply the nonlinear generalized Rayleigh quotient method [15,20] and introduce the following extreme value $\begin{aligned} λ_{0}^{T} = inf_{u \in W_{0, per}^{1, 2} (D^{T}) ∖ {0}} λ_{0}^{T} (u), \\ λ_{1 P}^{T} = inf_{u \in W_{0, per}^{1, 2} (D^{T}) ∖ {0}} λ_{1 P}^{T} (u) . \end{aligned}$ where $λ_{0}^{T} (\cdot)$ , $λ_{1 P}^{T} (\cdot)$ are nonlinear generalized Rayleigh quotients which are expressed by the exact formulas (3.2), (3.9), respectively. Furthermore, in Section 3 we show that $0 < λ_{1 P}^{T} < λ_{0}^{T} < + \infty$ .

Our first result is as follows
Theorem 1.2.
Let Ω be a bounded strictly star-shaped domain in $R^{N}$ with $C^{2}$ -manifold boundary $\partial Ω$ . Assume $0 < q < p < 1$ such that $N (1 - q) (1 - p) - 2 (1 + q) (1 + p) > 0$ and $0 < T < + \infty$ . Then there exists $λ^{} (T) \in [λ_{1 P}^{T}, λ_{0}^{T})$ such that there holds:*
For $λ \in [λ^{} (T), + \infty)$ , problem (* 1.1 )–( 1.4 ) possesses a periodic least energy solution $u_{λ}^{T}$ such that $\begin{matrix} \{\begin{array}{ll} Φ_{λ, T} (u_{λ}^{T}) > 0, & if λ \in [λ^{} (T), λ_{0}^{T}), \\ Φ_{λ, T} (u_{λ}^{T}) = 0, & if λ = λ_{0}^{T}, \\ Φ_{λ, T} (u_{λ}^{T}) < 0, & if λ \in (λ_{0}^{T}, + \infty) . \end{array} \end{matrix}$ Moreover, for all* $λ \in [λ^{} (T), + \infty)$ , $u_{λ}^{T}$ satisfies* $Φ_{λ, T}^{″} (u_{λ}^{T}) > 0$ and $u_{λ}^{T} ⩾ 0$ in $D^{T}$ with $u_{λ}^{T} \in C^{1, γ} (\overline{D^{T}}) \cap C^{2} (D^{T})$ for some $γ \in (0, 1)$ .

If $λ = λ^{} (T)$ , then problem (* 1.1 )–( 1.4 ) admits a nonnegative periodic least energy solution $u_{λ^{} (T)}^{T}$ which is compactly supported in* Ω.

If $λ \in (λ^{} (T), + \infty)$ , then there exists a nonnegative periodic least energy solution of (* 1.1 )–( 1.4 ) which is not compactly supported in Ω. Moreover, if $λ \in [λ_{0}^{T}, + \infty)$ , every least energy periodic solution of ( 1.1 )–( 1.4 ) is not compactly supported in Ω.

For any $λ < λ_{1 P}^{T}$ , problem ( 1.1 )–( 1.4 ) cannot have a weak solution.

Remark 1.2.
Note that the assumption $0 < q < p < 1$ and $N (1 - q) (1 - p) - 2 (1 + q) (1 + p) > 0$ implies $N ⩾ 3$ , for $N \in N$ .

In the following result, we derive some new properties of periodic least energy solutions according to the value of the parameter $T > 0$ .
Theorem 1.3.
Assume that the assumption of Theorem 1.2 is satisfied.
There exists $T_{1} > 0$ such that for each $T \in (0, T_{1})$ , any compactly supported in Ω periodic least energy solution $u_{λ^{} (T)}^{T}$ of (* 1.1 )–( 1.4 ) has no compact support in $D^{T}$ .

There exists $T_{0} > 0$ such that for each $T > T_{0}$ any compactly supported in Ω least energy solution $u_{λ^{} (T)}^{T}$ of (* 1.1 )–( 1.4 ) is periodically nontrivial, i.e., $\int_{D^{T}} | {(u_{λ^{*} (T)}^{T})}_{z} |^{2} d x d z \neq 0$ .

Regarding the outline of the paper: in Section 2, we present some preliminaries including functional space setting, the Pohozaev functional and the curve of critical exponents. In Section 3, we mention nonlinear generalized Rayleigh’s quotients and their extremal. Section 4 contains minimization arguments over the subset $M_{λ, T}$ of the Nehari manifold. In Sections 5 and 6 we prove Theorems 1.2 and 1.3, respectively. Section 7 is devoted to conclusion remarks and discussion of open problems. Afterwards, in the appendices we present some auxiliary results. In Appendix A, we prove some additional results concerning compactly supported solution to problem ( 1 R M ∗ ). We believe that the new approach used in this appendix (in particular see Proposition A.1) can be applied for other related problems. In Appendix B, we have some convergence result about the minimizers and finally in Appendix C, we prove a Pohozaev identity for solutions periodic in one variable.
2. Preliminaries

For $0 < T ⩽ + \infty$ , we use the standard notations: $L^{p} (D^{T})$ , $1 ⩽ p ⩽ + \infty$ , for the Lebesgue spaces, ${\overset{\circ}{W}}^{1, 2} (Ω)$ for the closure of $C_{0}^{\infty} (Ω)$ in the norm $‖ u ‖_{1} : = {(\int_{Ω} | \nabla_{x} u |^{2} d x)}^{1 / 2}$ . In what follows, $D_{0}^{T} : = C^{\infty} ([- T, T], C_{0}^{\infty} (Ω))$ , $0 < T ⩽ + \infty$ denotes the collection of functions $u (\cdot, \cdot)$ from $C^{\infty} ([- T, T] \times Ω)$ such that $u (z, \cdot) \in C_{0}^{\infty} (Ω)$ , $\forall z \in [- T, T]$ , and where we adopt the convention $[- T, T] = R$ for $T = \infty$ . For $0 < T ⩽ + \infty$ , we denote by $W_{0}^{1, 2} (D^{T})$ the closure of $D_{0}^{T}$ in the norm $‖ u ‖_{W^{1, 2} (D^{T})} = {(\int_{D^{T}} (| u |^{2} + | \nabla u |^{2}) d x d z)}^{1 / 2}$ .

It is not hard to show that for $0 < T ⩽ + \infty$ , any $u \in W_{0}^{1, 2} (D^{T})$ satisfies the Poincaré inequality, and thus $W_{0}^{1, 2} (D^{T})$ enjoys the equivalent norm $‖ u ‖_{W_{0}^{1, 2} (D^{T})} = ‖ \nabla u ‖_{L^{2} (D^{T})}$ . Notice that $W_{0}^{1, 2} (D^{T}) \subset C ([- T, T], W_{0}^{1, 2} (Ω))$ (see e.g., [29, Lemma 1.2]). Hence for $0 < T < + \infty$ and all $u \in W_{0}^{1, 2} (D^{T})$ , the periodicity condition $u (- T, \cdot) = u (T, \cdot)$ , is well defined and the set of functions $\begin{matrix} W_{0, per}^{1, 2} (D^{T}) : = {u \in W_{0}^{1, 2} (D^{T}) : u (- T, \cdot) = u (T, \cdot)} \end{matrix}$ defines a closed subspace in $W_{0}^{1, 2} (D^{T})$ . In what follows we denote $\begin{matrix} D_{0, per}^{T} : = {ψ \in C^{\infty} ([- T, T], C_{0}^{\infty} (Ω)) : ψ (- T, x) = ψ (T, x), x \in Ω} . \end{matrix}$ We call $u \in W_{0, per}^{1, 2} (D^{T})$ weak solution of (1.1)–(1.4) if $\begin{matrix} (2.1) & \int_{D^{T}} \nabla_{x} u \cdot \nabla_{x} ϕ d x d z + \int_{D^{T}} u_{z} \cdot ϕ_{z} d x d z = \int_{D^{T}} f_{λ} (u) ϕ d x d z, \forall ϕ \in D_{0, per}^{T} . \end{matrix}$ Observe that equality (2.1) for $u \in W_{0, per}^{1, 2} (D^{T})$ implies periodicity condition (1.3). Indeed, from (2.1) it can be easily shown that $\begin{aligned} 0 = & \int_{T}^{- T} [(u_{z} (z), ϕ_{z} (z)) - (Δ_{x} u (z), ϕ (z)) - (f_{λ} (u (z)), ϕ (z))] d z \\ = & \int_{T}^{- T} [- (u_{z z} (z), ϕ (z)) - (Δ_{x} u (z), ϕ (z)) - (f_{λ} (u (z)), ϕ (z))] d z \\ + (u_{z} (T), ϕ (T)) - (u_{z} (- T), ϕ (- T)), \forall ϕ \in D_{0, per}^{T} . \end{aligned}$ Here $(\cdot, \cdot)$ denotes the conjugation $W_{0}^{1, 2} (Ω)$ with its dual space. Hence, and since $ϕ (T) = ϕ (- T) \in C_{0}^{\infty} (Ω)$ we obtain that the equality $u_{z} (T) = u_{z} (- T)$ holds in the distribution sense.

Consider the Pohozaev functional $P_{λ, T} (u)$ , for $u \in W_{0}^{1, 2} (D^{T})$ . Then $\begin{matrix} (2.2) & Φ_{λ, T} (u) = P_{λ, T} (u) + \frac{1}{N} \int_{D^{T}} | \nabla_{x} u |^{2} d x d z, \forall u \in W_{0, per}^{1, 2} (D^{T}) . \end{matrix}$ We need a suitable Pohozaev identity in the spirit of [34] for periodic by z solutions of (1.1)–(1.4) (for the proof see Appendix C):

Lemma 2.1.
Assume that $\partial Ω$ is a $C^{2}$ -manifold and $N ⩾ 3$ . Let $u \in C^{1} (\overline{D^{T}}) \cap C^{2} (D^{T})$ be a solution of ( 1.1 )–( 1.4 ). Then there holds the Pohozaev identity $\begin{matrix} P_{λ, T} (u) = - \frac{1}{2 N} \int_{- T}^{T} \int_{\partial Ω} {| \frac{\partial u}{\partial ν} |}^{2} (x \cdot ν (x)) d σ (x) . \end{matrix}$ Here $d σ$ denotes the surface measure on $\partial Ω$ .
Proposition 2.1.
If $\frac{d}{d t} Φ_{λ, T} (t u) : = Φ_{λ, T}^{'} (t u) ⩽ 0$ for $u \neq 0$ , then $\frac{d}{d t} P_{λ, T} (t u) : = P_{λ, T}^{'} (t u) < 0$ .
Proof.
Note $P_{λ, T}^{'} (t u) = Φ_{λ, T}^{'} (t u) - \frac{2 t}{N} \int_{D^{T}} | \nabla_{x} u |^{2} d x d z$ . Since $Φ_{λ, T}^{'} (t u) ⩽ 0$ , this implies $P_{λ, T}^{'} (t u) ⩽ - (2 t / N) \int_{D^{T}} | \nabla_{x} u |^{2} d x d z < 0$ . □

Observe that if Ω is a star-shaped (resp. strictly star-shaped) domain with respect to the origin of coordinates of $R^{N}$ , then $x \cdot ν ⩾ 0$ (resp. $x \cdot ν > 0$ ) for all $x \in \partial Ω$ . This and Lemma 2.1 imply:
Corollary 2.1.
Let Ω be a bounded star-shaped domain in $R^{N}$ with a $C^{2}$ -manifold boundary $\partial Ω$ . Then any solution $u \in C^{1} (\overline{D^{T}}) \cap C^{2} (D^{T})$ of ( 1.1 )–( 1.4 ) satisfies $P_{λ, T} (u) ⩽ 0$ . Moreover, if $u (z, \cdot)$ has a compact support in Ω, then $P_{λ, T} (u) = 0$ . Furthermore, in the case where Ω is strictly star-shaped, the converse is also true: if $P_{λ} (u) = 0$ and $u \in C^{1} (\overline{D^{T}}) \cap C^{2} (D^{T})$ is a weak solution of ( 1.1 )–( 1.4 ), then u has a compact support in Ω.

For $u \in W_{0, per}^{1, 2} (D^{T}) ∖ {0}$ , following [21], we consider the system $\begin{matrix} (2.3) & \{\begin{array}{l} \int_{D^{T}} | \nabla u |^{2} d x d z - λ \int_{D^{T}} | u |^{p + 1} d x d z + \int_{D^{T}} | u |^{q + 1} d x d z = Φ_{λ, T}^{'} (u), \\ \int_{D^{T}} | \nabla u |^{2} d x d z - λ p \int_{D^{T}} | u |^{p + 1} d x d z + q \int_{D^{T}} | u |^{q + 1} d x d z = Φ_{λ, T}^{″} (u), \\ \frac{1}{2^{}} \int_{D^{T}} | \nabla u |^{2} d x d z - \frac{λ}{p + 1} \int_{D^{T}} | u |^{p + 1} d x d z + \frac{1}{q + 1} \int_{D^{T}} | u |^{q + 1} d x d z \\ = P_{λ, T} (u) - \frac{1}{N} \int_{D^{T}} | u_{z} |^{2} d x d z, \end{array} \end{matrix}$ where $\int_{D^{T}} | \nabla u |^{2} d x d z$ , $λ \int_{D^{T}} | u |^{p + 1} d x d z$ and $\int_{D^{T}} | u |^{q + 1} d x d z$ are considered as unknown values. A straighforward computation of the determinant leads to $\begin{matrix} d : = det (\begin{array}{ccc} 1 & 1 & 1 \\ 1 & p & q \\ \frac{1}{2^{}} & \frac{1}{p + 1} & \frac{1}{q + 1} \end{array}) = (q - p) (\frac{1}{2^{}} - \frac{p + q}{(p + 1) (q + 1)}) = \frac{(q - p)}{2 N (p + 1) (q + 1)} d^{} \end{matrix}$ where $d^{} : = d^{} (p, q, N) = N (1 - q) (1 - p) - 2 (1 + q) (1 + p)$ . Note that $(q, p) \in E_{s} (N)$ iff $d^{} > 0$ . Lemma 2.2.
Assume that* $(q, p) \in E_{s} (N)$ . Let $u \in W_{0, per}^{1, 2} (D^{T}) ∖ {0}$ be such that $P_{λ, T} (u) ⩽ 0$ and $Φ_{λ, T}^{'} (u) = 0$ , then $Φ_{λ, T}^{″} (u) > 0$ .
Proof.
Let $Φ_{λ, T}^{'} (u) = 0$ . Then since $d^{} > 0$ , we found from (2.3) that $\begin{matrix} \int_{D^{T}} | \nabla u |^{2} d x d z = \frac{2 N (p + 1) (q + 1)}{d^{}} Φ_{λ, T}^{″} (u) + \frac{(p + 1) (q + 1)}{d^{}} (P_{λ, T} (u) - \frac{1}{N} T_{z} (u)), \end{matrix}$ where $T_{z} (u) : = \int_{D^{T}} | u_{z} |^{2} d x d z$ . The Poincaré inequality entails $\int_{D^{T}} | \nabla u |^{2} d x d z > 0$ for $u \neq 0$ . Hence the inequalities $P_{λ, T} (u) - \frac{1}{N} T_{z} (u) ⩽ 0$ and $d^{} > 0$ imply that $Φ_{λ, T}^{″} (u) > 0$ . □

3. Nonlinear generalized Rayleigh’s quotients

This section is devoted to obtain the extreme values of the applicability of the Nehari manifold method for the parameter λ. To this end, we apply the nonlinear generalized Rayleigh quotient method (see [20]).

Let $T \in (0, + \infty]$ . We first consider the so-called zero energy level Rayleigh’s quotient (cf. [15]): $\begin{matrix} R_{T}^{0} (u) = \frac{\frac{1}{2} \int_{D^{T}} | \nabla u |^{2} d x d z + \frac{1}{q + 1} \int_{D^{T}} | u |^{q + 1} d x d z}{\frac{1}{p + 1} \int_{D^{T}} | u |^{p + 1} d x d z}, u \in W_{0}^{1, 2} (D^{T}) ∖ {0} . \end{matrix}$ Note that for any $u \in W_{0}^{1, 2} (D^{T}) ∖ {0}$ and $λ \in R$ , $\begin{matrix} (3.1) & if R_{T}^{0} (u) = λ, then Φ_{λ, T} (u) = 0 . \end{matrix}$ It is easy to see that $\frac{d}{d t} R_{T}^{0} (t u) = 0$ if and only if $\begin{matrix} (1 - p) \frac{t^{- p}}{2} \int_{D^{T}} | \nabla u |^{2} d x d z + (q - p) \frac{t^{q - p - 1}}{q + 1} \int_{D^{T}} | u |^{q + 1} d x d z = 0 \end{matrix}$ and that the only solution to this equation is $\begin{matrix} t_{0} (u) = {(\frac{2 (p - q)}{(1 - p) (q + 1)} \frac{\int_{D^{T}} | u |^{q + 1} d x d z}{\int_{D^{T}} | \nabla u |^{2} d x d z})}^{\frac{1}{1 - q}} . \end{matrix}$ Note that $t_{0} (u)$ is a value where the function $R_{T}^{0} (t u)$ attains its global minimum. Substituting $t_{0} (u)$ into $R_{T}^{0} (t)$ we obtain the nonlinear generalized Rayleigh quotient: $\begin{matrix} (3.2) & λ_{0}^{T} (u) = R_{T}^{0} (t_{0} (u) u) = c_{0}^{q, p} \frac{{(\int_{D^{T}} | \nabla u |^{2} d x d z)}^{\frac{p - q}{1 - q}} {(\int_{D^{T}} | u |^{q + 1} d x d z)}^{\frac{1 - p}{1 - q}}}{\int_{D^{T}} | u |^{p + 1} d x d z}, \end{matrix}$ where $\begin{matrix} (3.3) & c_{0}^{q, p} = \frac{(1 - q) (p + 1)}{(1 - p) (q + 1)} {(\frac{(1 - p) (q + 1)}{2 (p - q)})}^{\frac{p - q}{1 - q}} . \end{matrix}$ It is not difficult to check that $\begin{array}{r} (3.4) & Φ_{λ_{0}^{T} (u), T} (t_{0} (u) u) = 0, Φ_{λ_{0}^{T} (u), T}^{'} (t_{0} (u) u) = 0, \end{array}$ and that the map $λ_{0}^{T} (\cdot) : W_{0}^{1, 2} (D^{T}) ∖ 0 \to R$ is a $C^{1}$ -functional. For $T \in (0, + \infty]$ , consider $\begin{matrix} (3.5) & λ_{0}^{T} = inf_{u \in W_{0, per}^{1, 2} (D^{T}) ∖ {0}} λ_{0}^{T} (u) . \end{matrix}$ Using Sobolev’s and Holder’s inequalities (see e.g., [21]) it can be shown that $\begin{matrix} 0 < λ_{0}^{T} < + \infty for all T \in (0, + \infty) . \end{matrix}$ It is important that (3.1) and (3.5) imply the following

Proposition 3.1.
For $T \in (0, + \infty)$
If $λ < λ_{0}^{T}$ , then $Φ_{λ, T} (u) > 0$ for any $u \neq 0$ ,

If $λ > λ_{0}^{T}$ , then there is $u \in W_{0, per}^{1, 2} (D^{T}) ∖ {0}$ such that $Φ_{λ, T} (u) < 0$ .

Furthermore, we have
Lemma 3.1.
For any $T \in (0, + \infty)$ , there exists a minimizer ${\hat{u}}_{0}^{T} \in W_{0, per}^{1, 2} (D^{T}) ∖ {0}$ of ( 3.5 ). Moreover, $D Φ_{λ_{0}^{T}, T} ({\hat{u}}_{0}^{T}) = 0$ and $Φ_{λ_{0}^{T}, T} ({\hat{u}}_{0}^{T}) = 0$ .
Proof.
Let $T \in (0, + \infty)$ and let ${(u_{n})}_{n = 1}^{\infty}$ be a minimizing sequence of (3.5). Due to the homogeneity of $λ_{0}^{T} (u)$ , one can find constants $a_{n}$ such that $λ_{0}^{T} (a_{n} u_{n}) = λ_{0}^{T} (u_{n})$ and $‖ v_{n} ‖_{1} = 1$ , where $v_{n} = a_{n} u_{n}$ for $n \in N$ . Hence $(v_{n})$ is a minimizing sequence of (3.5) which is bounded in $W_{0, per}^{1, 2} (D^{T})$ . Thus there exists a subsequence, again denoted by $(v_{n})$ , such that $v_{n} ⇀ {\hat{v}}_{0}$ weakly in $W_{0, per}^{1, 2} (D^{T})$ and strongly $v_{n} \to {\hat{v}}_{0}$ in $L^{γ} (D^{T})$ , for $1 < γ < 2_{N + 1}^{}$ and for some ${\hat{v}}_{0} \in W_{0, per}^{1, 2} (D^{T})$ . Note that ${\hat{v}}_{0} \neq 0$ . Indeed, if this is not true, then we have $\begin{aligned} (3.6) & λ_{0}^{T} (v_{n}) = c_{0}^{q, p} \frac{{(\int_{D^{T}} | v_{n} |^{q + 1} d x d z)}^{\frac{1 - p}{1 - q}}}{\int_{D^{T}} | v_{n} |^{p + 1} d x d z} ⩾ C {(\int_{D^{T}} | v_{n} |^{q + 1} d x d z)}^{- \frac{(2_{N}^{} - 2) (p - q)}{(2_{N}^{} - (1 + q)) (1 - q)}} \to + \infty, \end{aligned}$ since by Sobolev’s, Poincaré’s and Holder’s inequalities, we obtain $\begin{aligned} \int_{D^{T}} | v_{n} |^{p + 1} d x d z & ⩽ {(\int_{D^{T}} | v_{n} |^{q + 1} d x d z)}^{\frac{γ}{1 + q}} {(\int_{D^{T}} | v_{n} |^{2_{N}^{}} d x d z)}^{\frac{1 + p - γ}{2_{N}^{}}} \\ (3.7) & ⩽ C {(\int_{D^{T}} | v_{n} |^{q + 1} d x d z)}^{\frac{γ}{1 + q}}, \end{aligned}$ where $γ = \frac{(2_{N}^{} - (1 + p)) (1 + q)}{2_{N}^{} - (1 + q)}$ and $0 < C < + \infty$ does not depend on $n \in N$ . Observe that $λ_{0}^{T} (\cdot)$ is weakly lower semicontinuous and bounded below functional on $W_{0, per}^{1, 2} (D^{T})$ . This easily yields that ${\hat{v}}_{0} \in W_{0, per}^{1, 2} (D^{T}) ∖ {0}$ is a minimizer of $λ_{0}^{T} (u)$ . Hence $\begin{matrix} D λ_{0}^{T} ({\hat{v}}_{0}) (ϕ) = t_{O} ({\hat{v}}_{0}) D R_{T}^{0} (t {\hat{v}}_{0}) |_{t = t_{0} ({\hat{v}}_{0})} (ϕ) + \frac{\partial}{\partial t} R_{T}^{0} (t_{0} ({\hat{v}}_{0}) {\hat{v}}_{0}) (D t_{0} ({\hat{v}}_{0}) (ϕ)) = 0, \forall ϕ \in D_{0, per}^{T} . \end{matrix}$ Since $\partial R_{T}^{0} (t_{0} ({\hat{v}}_{0}) {\hat{v}}_{0}) / \partial t = 0$ , this implies that $\begin{matrix} D R_{T}^{0} (u) |_{u = t_{0} ({\hat{v}}_{0}) {\hat{v}}_{0}} (ϕ) = 0, \forall ϕ \in D_{0, per}^{T} . \end{matrix}$ Denote ${\hat{u}}_{0}^{T} = t_{0} ({\hat{v}}_{0}) {\hat{v}}_{0}$ . Then due to the equality $λ_{0}^{T} = R_{T}^{0} ({\hat{u}}_{0}^{T})$ we have $\begin{matrix} 0 = D R_{T}^{0} ({\hat{u}}_{0}^{T}) = \frac{1}{\int_{D^{T}} | u |^{p + 1} d x d z} \cdot D Φ_{λ_{0}^{T}, T} ({\hat{u}}_{0}^{T}), \end{matrix}$ which yields that $D Φ_{λ_{0}^{T}, T} ({\hat{u}}_{0}^{T}) = 0$ and since $λ_{0}^{T} = R_{T}^{0} ({\hat{u}}_{0}^{T})$ , $Φ_{λ_{0}^{T}, T} ({\hat{u}}_{0}^{T}) = 0$ . □

Let $0 < T ⩽ + \infty$ . We shall also need the following Rayleigh’s quotients: $\begin{aligned} R_{T}^{P} (u) = \frac{\frac{1}{2_{N}^{}} \int_{D^{T}} | \nabla_{x} u |^{2} d x d z + \frac{1}{2} \int_{D^{T}} | u_{z} |^{2} d x d z + \frac{1}{q + 1} \int_{D^{T}} | u |^{q + 1} d x d z}{\frac{1}{p + 1} \int_{D^{T}} | u |^{p + 1} d x d z}, \\ R_{T}^{1} (u) = \frac{\int_{D^{T}} | \nabla_{x} u |^{2} d x d z + \int_{D^{T}} | u_{z} |^{2} d x d z + \int_{D^{T}} | u |^{q + 1} d x d z}{\int_{D^{T}} | u |^{p + 1} d x d z}, u \neq 0 . \end{aligned}$ Notice that for any $u \neq 0$ and $λ \in R$ , $\begin{matrix} (3.8) & R_{T}^{P} (u) = λ \Leftrightarrow P_{λ, T} (u) = 0 and R_{T}^{1} (u) = λ \Leftrightarrow Φ_{λ, T}^{'} (u) = 0 . \end{matrix}$ Arguing as above for $R_{T}^{0} (t u)$ , it can be shown that each of functions $R_{T}^{P} (t u)$ and $R_{T}^{1} (t u)$ attains its global minimum at some point, $t_{P} (u)$ and $t_{1} (u)$ , respectively.

Moreover, it is easily seen that the following equation $\begin{matrix} R_{T}^{P} (t u) = R_{T}^{1} (t u), t > 0, \end{matrix}$ has a unique solution $\begin{matrix} t_{1 P} (u) = {(\frac{(p - q)}{(q + 1)} \frac{\int_{D^{T}} | u |^{q + 1} d x d z}{\frac{(2_{N}^{} - p - 1)}{2_{N}^{}} \int_{D^{T}} | \nabla_{x} u |^{2} d x d z + \frac{1 - p}{2} \int_{D^{T}} | u_{z} |^{2} d x d z})}^{\frac{1}{1 - q}} . \end{matrix}$ Thus, for $0 < T ⩽ + \infty$ , we are able to introduce the following nonlinear generalized Rayleigh quotient $\begin{matrix} (3.9) & \begin{aligned} λ_{1 P}^{T} (u) & : = R_{T}^{P} (t_{1 P} (u) u) \\ = R_{T}^{1} (t_{1 P} (u) u) \\ = c_{1 P}^{q, p} \frac{(\frac{2_{N}^{} - 1 - q}{2_{N}^{}} \int_{D^{T}} | \nabla_{x} u |^{2} d x d z + \frac{1 - q}{2} \int_{D^{T}} | u_{z} |^{2} d x d z) {(\int_{D^{T}} | u |^{q + 1} d x d z)}^{\frac{1 - p}{1 - q}}}{(\int_{D^{T}} | u |^{p + 1} d x d z) {(\frac{(2_{N}^{} - p - 1)}{2_{N}^{}} \int_{D^{T}} | \nabla_{x} u |^{2} d x d z + \frac{1 - p}{2} \int_{D^{T}} | u_{z} |^{2} d x d z)}^{\frac{1 - p}{1 - q}}}, \end{aligned} \end{matrix}$ where $\begin{matrix} c_{1 P}^{q, p} = \frac{p + 1}{p - q} {(\frac{p - q}{1 + q})}^{\frac{1 - p}{1 - q}} . \end{matrix}$ It is easily seen that $λ_{1 P}^{T} (\cdot) \in C^{1} (W_{0}^{1, 2} (D^{T}) ∖ 0)$ . Notice that $\begin{matrix} P_{λ_{1 P}^{T} (u)} (t_{1 P} (u) u) = 0 and Φ_{λ_{1 P} (u), T}^{'} (t_{1 P} (u) u) = 0, \forall u \neq 0 . \end{matrix}$ Recall that by Corollary 2.1, if $u \in C^{1} (\overline{D^{T}}) \cap C^{2} (D^{T})$ is a compactly supported solution of (1.1), then $\begin{matrix} P_{λ, T} (u) = 0, Φ_{λ, T}^{'} (u) = 0 \end{matrix}$ which implies that $t_{1 P} (u) = 1$ and $λ = λ_{1 P}^{T} (u)$ . Consider $\begin{matrix} (3.10) & λ_{1 P}^{T} = inf_{u \in W_{0, per}^{1, 2} (D^{T}) ∖ {0}} λ_{1 P}^{T} (u), T \in (0, + \infty] . \end{matrix}$ Using Sobolev’s and Hölder’s inequalities, similar to (3.6), it can be shown that $\begin{matrix} 0 < λ_{1 P}^{T} < + \infty, 0 < T ⩽ \infty . \end{matrix}$
Lemma 3.2.
Let $T \in (0, + \infty)$ . For any $u \in W_{0, per}^{1, 2} (D^{T}) ∖ {0}$ ,
$R_{T}^{P} (t u) > R_{T}^{1} (t u)$ iff $t \in (0, t_{1 P} (u))$ and $R_{T}^{P} (t u) < R_{T}^{1} (t u)$ iff $t \in (t_{1 P} (u), + \infty)$ ;

for $(q, p) \in E_{s} (N)$ , there holds $t_{1} (u) < t_{1 P} (u) < t_{P} (u)$ ;

for $(q, p) \in E_{s} (N)$ , $t_{1} (u) < t_{1 P} (u) < t_{0} (u)$ .

Proof.
Observe that $R_{T}^{P} (t u) / R_{T}^{1} (t u) \to \frac{p + 1}{q + 1} > 1$ as $t \to 0$ . Hence, from the uniqueness of $t_{1 P} (u)$ we obtain (i).

By (3.8) we have $Φ_{λ_{1 P}^{T} (u), T}^{'} (u) = 0$ . Therefore Proposition 2.1 implies $\frac{d}{d t} P_{λ_{1 P} (u), T} (t_{1 P} (u) u) < 0$ . Hence and since $\begin{matrix} \frac{d}{d t} R_{T}^{P} (t u) |_{t = t_{1 P} (u)} = \frac{p + 1}{\int_{D^{T}} | t u |^{p + 1} d x d z} \cdot \frac{d}{d t} P_{λ_{1 P}^{T} (u), T} (t u) |_{t = t_{1 P} (u)}, \end{matrix}$ we conclude that $\frac{d}{d t} R_{T}^{P} (t u) |_{t = t_{1 P} (u)} < 0$ . Now taking into account that $t_{P} (u)$ is a point of global minimum of $R_{T}^{P} (t u)$ we obtain that $t_{1 P} (u) < t_{P} (u)$ . To prove $t_{1} (u) < t_{1 P} (u)$ , first observe that $\begin{matrix} \frac{d}{d t} R_{T}^{1} (t u) |_{t = t_{1 P} (u)} = \frac{t}{\int_{D^{T}} | t u |^{p + 1} d x d z} \cdot Φ_{λ_{1 P}^{T} (u), T}^{″} (t u) |_{t = t_{1 P} (u)}, \end{matrix}$ and by Lemma 2.2 the equalities $Φ_{λ_{1 P}^{T} (u), T}^{'} (t_{1 P} (u) u) = 0$ and $P_{λ_{1 P}^{T} (u), T} (t_{1 P} (u) u) = 0$ imply that $Φ_{λ_{1 P}^{T} (u), T}^{″} (t_{1 P} (u) u) > 0$ . Thus $\frac{d}{d t} R_{T}^{1} (t_{1 P} (u) u) > 0$ and the proof of (ii) follows.

Note that $\begin{matrix} {(R_{T}^{0})}^{'} (t u) = \frac{(p + 1)}{t} (R_{T}^{1} (t u) - R_{T}^{0} (t u)) . \end{matrix}$ Hence, $\begin{matrix} (3.11) & {(R_{T}^{0})}^{'} (t u) = 0 \Leftrightarrow R_{T}^{1} (t u) = R_{T}^{0} (t u) \end{matrix}$ is satisfied by $t = t_{0} (u)$ . Furthermore, if ${(R_{T}^{0})}^{'} (t u) = 0$ , then $\begin{matrix} 0 < {(R_{T}^{0})}^{″} (t u) = \frac{p + 1}{t} ({(R_{T}^{1})}^{'} (t u) - \frac{p + 2}{p + 1} {(R_{T}^{0})}^{'} (t u)) = \frac{p + 1}{t} {(R_{T}^{1})}^{'} (t u) . \end{matrix}$ From this and part (i) (note that $R_{T}^{P} (t_{0} (u) u) < R_{T}^{0} (t_{0} (u) u) = R_{T}^{1} (t_{0} (u) u)$ ) we get the conclusion of part (iii). □

Now we can prove the following main property of the extreme value $λ_{1 P}^{T}$ : Lemma 3.3.
Assume $(q, p) \in E_{s} (N)$ .
If $λ < λ_{1 P}^{T}$ , $Φ_{λ, T}^{'} (u) = 0$ and $u ≢ 0$ , then $P_{λ, T} (u) > 0$ .

For any $λ > λ_{1 P}^{T}$ , there exists $u \in W_{0}^{1, 2} (D^{T}) ∖ {0}$ such that $Φ_{λ, T}^{'} (u) = 0$ and $P_{λ, T} (u) < 0$ .

Proof.
Let us prove (i). Since $Φ_{λ, T}^{'} (u) = 0$ , then $R_{T}^{1} (u) = λ$ . Arguing by contradiction, suppose $P_{λ, T} (u) ⩽ 0$ . Then $R_{T}^{P} (u) ⩽ λ = R_{T}^{1} (u)$ , and therefore by Lemma 3.2, $1 ⩾ t_{1, P} (u) > t_{1} (u)$ . Note that ${(R_{T}^{1})}^{'} (t u) > 0$ for $t > t_{1} (u)$ . Hence $R_{T}^{1} (u) ⩾ λ_{1 P}^{T} (u) > λ$ from which we get a contradiction.

We now prove (ii). From $λ > λ_{1 P}^{T}$ , we deduce that there exists $v \in W_{0}^{1, 2} (D^{T}) ∖ {0}$ such that $λ > λ_{1 P}^{T} (v) = R_{T}^{1} (t_{1, P} (v) v)$ . Therefore there exists $θ > t_{1 P} (v)$ such that $R_{T}^{1} (θ v) = λ$ . We have also from Lemma 3.2 that $R_{T}^{P} (θ v) < R_{T}^{1} (θ v) = λ$ from which we get $P_{λ, T} (θ v) < 0$ . Setting $u = θ v$ we complete the proof of assertion (ii). □
Corollary 3.1.
There holds $λ_{1 P}^{T} < λ_{0}^{T}$ .
Proof.
By Lemma 3.2(iii), for any $u \in W_{0}^{1, 2} (D^{T}) ∖ {0}$ , $t_{1} (u) < t_{1, P} (u) < t_{0} (u)$ . In addition, for any $u \in W_{0}^{1, 2} (D^{T}) ∖ {0}$ , by (3.11) we have $R_{T}^{0} (t_{0} (u) u) = R_{T}^{1} (t_{0} (u) u)$ . Therefore, since ${(R_{T}^{1})}^{'} (t u) > 0$ for $t \in (t_{1} (u), t_{0} (u))$ , and while $t_{1 P} (u) \in (t_{1} (u), t_{0} (u))$ , we have $\begin{matrix} λ_{0}^{T} (u) = R_{T}^{0} (t_{0} (u) u) = R_{T}^{1} (t_{0} (u) u) > R_{T}^{1} (t_{1 P} (u) u) = R_{T}^{P} (t_{1 P} (u) u) . \end{matrix}$ By Lemma 3.1, there exists a minimizer ${\hat{u}}_{0}^{T} \in W_{0, per}^{1, 2} (D^{T}) ∖ {0}$ of $λ_{0}^{T} (u)$ . We thus have $\begin{matrix} λ_{0}^{T} = R_{T}^{0} (t_{0} ({\hat{u}}_{0}^{T}) {\hat{u}}_{0}^{T}) > R_{T}^{P} (t_{1 P} ({\hat{u}}_{0}^{T}) {\hat{u}}_{0}^{T}) ⩾ λ_{1 P}^{T} . \end{matrix}$ □
Corollary 3.2.
Assume $(q, p) \in E_{s} (N)$ . Let Ω be a bounded star-shaped domain in $R^{N}$ with $C^{2}$ -manifold boundary $\partial Ω$ . Then for any $λ < λ_{1 P}^{T}$ problem ( 1.1 )–( 1.4 ) cannot have a weak solution.
Proof.
We argue by contradiction. Suppose that there exists a non zero weak solution $u \in W_{0, per}^{1, 2} (D^{T})$ of (1.1)–(1.4) for some $λ < λ_{1 P}^{T}$ . Then $Φ_{λ, T}^{'} (u) = 0$ . By the regularity solutions of elliptic problems (see [17]) it follows that $u \in C^{1, γ} (\overline{D^{T}}) \cap C^{2} (D^{T})$ for some $γ \in (0, 1)$ . Hence, Lemma 3.3 implies that $P_{λ, T} (u) > 0$ which yields a contradiction on account of Corollary 2.1. □

4. Minimization problem with Pohozaev’s function as a constraint

Let $T \in (0, \infty)$ and let $λ ⩾ λ_{1 P}^{T}$ . Recall that $\begin{matrix} M_{λ, T} : = {u \in W_{0, per}^{1, 2} (D^{T}) ∖ {0} : Φ_{λ, T}^{'} (u) = 0, P_{λ, T} (u) ⩽ 0} . \end{matrix}$ Consider the following constrained minimization problem: $\begin{matrix} (4.1) & {\hat{Φ}}_{λ, T} : = min_{u \in M_{λ, T}} Φ_{λ, T} (u) . \end{matrix}$

Proposition 4.1.
Assume $(q, p) \in E_{s} (N)$ . $M_{λ, T} \neq \emptyset$ if and only if $λ ⩾ λ_{1 P}^{T}$ .
Proof.
Let $λ < λ_{1 P}^{T}$ and $u \in W_{0, per}^{1, 2} (D^{T}) ∖ {0}$ such that $Φ_{λ, T}^{'} (u) = 0$ . Then by Lemma 3.3, $P_{λ, T} (u) > 0$ . Hence $M_{λ, T} = \emptyset$ for any $λ < λ_{1 P}^{T}$ .

Let $λ > λ_{1 P}^{T}$ . By Lemma 3.3, there exists $u \in W_{0, per}^{1, 2} (D^{T}) ∖ {0}$ such that $Φ_{λ, T}^{'} (u) = 0$ and $P_{λ, T} (u) < 0$ , and therefore $u \in M_{λ, T}$ .

Now consider the case $λ = λ_{1 P}^{T}$ . Let ${(λ_{n})}_{n \in N}$ be a sequence such that $λ_{n} ↓ λ_{1 P}^{T}$ as $n \to \infty$ . Thus, there exists ${(u_{n})}_{n \in N} \subset W_{0, per}^{1, 2} (D^{T}) ∖ {0}$ satisfying $P_{λ_{n}, T} (u_{n}) ⩽ 0 = Φ_{λ_{n}, T}^{'} (u_{n})$ . Arguing as in [21, Lemma 9], it is not difficult to show that $(u_{n})$ is bounded in $W_{0, per}^{1, 2} (D^{T})$ and up to a subsequence $u_{n} ⇀ \tilde{u}$ weakly in $W_{0, per}^{1, 2} (D^{T}) ∖ {0}$ . Therefore passing to the limit as $n \to \infty$ , we get $Φ_{λ_{1 P}^{T}, T}^{'} (\tilde{u}) ⩽ 0$ and $P_{λ_{1 P}^{T}, T} (\tilde{u}) ⩽ 0$ . We next claim that $\tilde{u} ≢ 0$ . Set $u_{n} = t_{n} v_{n}$ with $‖ v_{n} ‖_{1} = 1$ . On the contrary, suppose that $t_{n} \to 0$ and or $v_{n} \to 0$ weakly, as $n \to \infty$ . In light of the relation $P_{λ_{n}, T} (u_{n}) ⩽ 0 = Φ_{λ_{n}, T}^{'} (u_{n})$ , we have $\begin{aligned} (4.2) & 1 + t_{n}^{q - 1} B_{n} = λ_{n} t_{n}^{p - 1} A_{n} and \frac{p - q}{q + 1} t_{n}^{q - 1} B_{n} ⩽ \frac{2_{N}^{} - p - 1}{2_{N}^{}}, \end{aligned}$ where we have set $A_{n} : = \int_{D^{T}} | v_{n} |^{p + 1} d x d z$ and $B_{n} : = \int_{D^{T}} | v_{n} |^{q + 1} d x d z$ . Assume that $t_{n}^{q - 1} B_{n} \to l$ , where $0 ⩽ l < \infty$ . Then, from (4.2), we conclude that $λ_{n} t_{n}^{p - 1} A_{n} \to 1 + l$ . Similar to (3.7), we have $\begin{array}{r} A_{n} ⩽ C B_{n}^{\frac{γ}{q + 1}}, \end{array}$ where $γ = \frac{(2_{N}^{} - (1 + p)) (1 + q)}{2_{N}^{} - (1 + q)}$ and $0 < C < \infty$ is independent of n. Consequently, $\begin{aligned} (4.3) & t_{n}^{p - 1} A_{n} ⩽ C t_{n}^{p - 1 - \frac{γ (q - 1)}{q + 1}} {(t_{n}^{q - 1} B_{n})}^{\frac{γ}{q + 1}} . \end{aligned}$ Since $p - 1 - \frac{γ (q - 1)}{q + 1} > 0$ , the right-hand side quantity of the above expression converges to 0, which yields a contradiction to the assumption $λ_{n} t_{n}^{p - 1} A_{n} \to 1 + l$ . The case $l = \infty$ can be also easily ruled out from (4.2) and (4.3). We now redo the same argument as for $λ > λ_{1 P}^{T}$ to get $M_{λ_{1 P}^{T}, T} \neq \emptyset$ . □

From here it follows that ${\hat{Φ}}_{λ, T} < + \infty$ for any $λ ⩾ λ_{1 P}^{T}$ . Lemma 4.1.
For any $λ > λ_{0}^{T}$ , there holds ${\hat{Φ}}_{λ, T} < 0$ .
Proof.
Since $λ > λ_{0}^{T}$ , there exists $v \in W_{0, per}^{1, 2} (D^{T}) ∖ {0}$ such that $λ > λ_{0}^{T} (v) ⩾ λ_{0}^{T}$ . Moreover, from (3.4), we have $Φ_{λ_{0}^{T} (v), T} (t_{0} (v) v) = 0 = Φ_{λ_{0}^{T} (v), T}^{'} (t_{0} (v) v)$ . Therefore, $Φ_{λ, T}^{'} (t_{0} (v) v) < 0$ . Then, there exist $t^{}$ and $\tilde{t}$ verifying $0 < t^{} < t_{0} (v) < \tilde{t}$ , $Φ_{λ, T}^{'} (t^{} v) = Φ_{λ, T}^{'} (\tilde{t} v) = 0$ and such that for any $t \in (t^{}, \tilde{t})$ one has $Φ_{λ, T}^{'} (t v) < 0$ . Thus $P_{λ, T} (t^{} v) ⩽ Φ_{λ, T} (t^{} v) ⩽ Φ_{λ, T} (t_{0} (v) v) < 0$ . Consequently, $t^{} v \in M_{λ, T}$ and ${\hat{Φ}}_{λ, T} = Φ_{λ, T} (u_{λ}) < 0$ . □
Lemma 4.2.
For any* $λ ⩾ λ_{1 P}^{T}$ , there exists a minimizer $u_{λ} \in M_{λ, T}$ of problem ( 4.1 ).
Proof.
Let $λ ⩾ λ_{1 P}^{T}$ . Then by the above $M_{λ, T} \neq \emptyset$ and ${\hat{Φ}}_{λ, T} < + \infty$ . Observe that $M_{λ, T}$ is bounded. Indeed, if $u \in M_{λ, T}$ , then $\begin{aligned} c_{1} ‖ u ‖_{1}^{2} & ⩽ \frac{1}{2_{N}^{}} \int_{D^{T}} | \nabla_{x} u |^{2} d x d z + \frac{1}{2} \int_{D^{T}} | u_{z} |^{2} d x d z + \frac{1}{q + 1} \int_{D^{T}} | u |^{q + 1} d x d z \\ ⩽ λ \frac{1}{p + 1} \int_{D^{T}} | u |^{p + 1} d x d z ⩽ c_{2} λ \frac{1}{p + 1} ‖ u ‖_{1}^{p + 1} \end{aligned}$ with some constants $c_{1}$ , $c_{2}$ which do not depend on $u \in M_{λ, T}$ . Hence, since $p + 1 < 2$ , we have $‖ u ‖_{1} ⩽ C < + \infty$ , $\forall u \in M_{λ, T}$ , where $C < + \infty$ does not depend on $u \in M_{λ, T}$ .

Let $(u_{m})$ be a minimizing sequence of (4.1): $\begin{matrix} Φ_{λ, T} (u_{m}) \to {\hat{Φ}}_{λ, T} as m \to \infty and u_{m} \in M_{λ, T}, m \in N . \end{matrix}$ Since $(u_{m})$ is bounded, there exists a subsequence, which we denote again $(u_{m})$ , such that $u_{m} ⇀ u_{λ}$ weakly in $W_{0, per}^{1, 2} (D^{T})$ and strongly $u_{m} \to u_{λ}$ in $L^{γ} (D^{T})$ , $1 < γ < 2_{N}^{}$ for some $u_{λ} \in W_{0, per}^{1, 2} (D^{T})$ . We claim that $u_{m} \to u_{λ}$ strongly in $W_{0, per}^{1, 2} (D^{T})$ . If not, $‖ u_{λ} ‖_{1} < {lim inf}_{m \to \infty} ‖ u_{m} ‖_{1}$ and this implies that $\begin{array}{r} Φ_{λ, T}^{'} (u_{λ}) < \underset{m \to \infty}{lim inf} Φ_{λ, T}^{'} (u_{m}) = 0 . \end{array}$ Hence $Φ_{λ, T}^{'} (u_{λ}) < 0$ and $u_{λ} \neq 0$ . Then there exists $κ > 1$ such that $Φ_{λ, T}^{'} (κ u_{λ}) = 0$ and $Φ_{λ, T} (κ u_{λ}) < Φ_{λ, T} (u_{λ}) < {\hat{Φ}}_{λ, T}$ . By Proposition 2.1, $Φ_{λ, T}^{'} (κ u_{λ}) = 0$ implies $P_{λ}^{'} (κ u_{λ}) < 0$ . From this and since $\begin{matrix} P_{λ} (u_{λ}) < \underset{m \to \infty}{lim inf} P_{λ} (u_{m}) ⩽ 0, \end{matrix}$ we conclude that $P_{λ} (κ u_{λ}) < 0$ . Thus $κ u_{λ} \in M_{λ, T}$ and $Φ_{λ, T} (κ u_{λ}) < {\hat{Φ}}_{λ, T}$ , which is a contradiction. Hence, $u_{m} \to u_{λ}$ strongly in $W_{0, per}^{1, 2} (D^{T})$ . The property that ${\hat{Φ}}_{λ, T} > 0$ for $λ \in (λ_{1 P}^{T}, λ_{0}^{T})$ , and ${\hat{Φ}}_{λ, T} < 0$ for $λ \in (λ_{0}^{T}, + \infty)$ yield that $u_{λ} \neq 0$ , for $λ \neq λ_{0}^{T}$ . In the case $λ = λ_{0}^{T}$ , the conclusion $u_{λ} \neq 0$ follows by the same arguments as in the proof of Lemma 3.1. Thus we get that $Φ_{λ, T} (u_{λ}) = {\hat{Φ}}_{λ, T}$ and $u_{λ} \in M_{λ, T}$ . □

5. Proof of Theorem 1.2

Let $λ ⩾ λ_{1 P}^{T}$ . By Lemma 4.2 there exists a minimizer $u_{λ} \in W_{0, per}^{1, 2} (D^{T}) ∖ {0}$ of (4.1). This implies that there exist Lagrange multipliers $μ_{0}$ , $μ_{1}$ , $μ_{2}$ such that $| μ_{0} | + | μ_{1} | + | μ_{2} | \neq 0$ , $μ_{0}$ and $μ_{2} ⩾ 0$ , and $\begin{array}{c} (5.1) & μ_{0} D Φ_{λ, T} (u_{λ}) + μ_{1} D Φ_{λ, T}^{'} (u_{λ}) + μ_{2} D P_{λ, T} (u_{λ}) = 0, \\ (5.2) & μ_{2} P_{λ, T} (u_{λ}) = 0 . \end{array}$

Proposition 5.1.
Assume $0 < q < p < 1$ and $(q, p) \in E_{s} (N)$ . Let $λ > λ_{1 P}^{T}$ and $u_{λ} \in W_{0, per}^{1, 2} (D^{T})$ be a minimizer in ( 4.1 ) such that $P_{λ, T} (u_{λ}) < 0$ . Then $u_{λ}$ is a weak solution of ( 1.1 )–( 1.4 ). Furthermore, one may assume that $u_{λ} ⩾ 0$ in $D^{T}$ .
Proof.
Since $P_{λ, T} (u_{λ}) < 0$ , equality (5.2) implies $μ_{2} = 0$ . Moreover, by Lemma 2.2 we have $Φ_{λ, T}^{″} (u_{λ}) > 0$ . Testing (5.1) by $u_{λ}$ we get $0 = μ_{0} Φ_{λ, T}^{'} (u_{λ}) + μ_{1} Φ_{λ, T}^{″} (u_{λ})$ , and consequently $μ_{1} Φ_{λ, T}^{″} (u_{λ}) = 0$ . Since $Φ_{λ, T}^{″} (u_{λ}) > 0$ , we get $μ_{1} = 0$ , consequently $μ_{0} \neq 0$ . Thus $D Φ_{λ, T} (u_{λ}) = 0$ , i.e., $u_{λ}$ is a weak solution of (1.1)–(1.4). Since $Φ_{λ, T} (| u |) = Φ_{λ, T} (u)$ , $Φ_{λ, T}^{'} (| u |) = Φ_{λ, T}^{'} (u)$ , $P_{λ, T} (| u |) = P_{λ, T} (u)$ for any $u \in W_{0, per}^{1, 2} (D^{T})$ we may assume that $u_{λ} ⩾ 0$ . □

Denote $\begin{matrix} G_{λ}^{T} = {u \in M_{λ, T} : Φ_{λ, T} (u_{λ}) = {\hat{Φ}}_{λ, T}} . \end{matrix}$ Select a subset in $G_{λ}^{T}$ which does not contain “isolated points”: $\begin{matrix} (5.3) & {\tilde{G}}_{λ}^{T} = {u_{λ} \in G_{λ}^{T} : \exists u_{λ_{m}} \in G_{λ_{m}}^{T}, λ_{m} > λ, m \in N s.t. u_{λ_{m}} \to u_{λ} in W_{0, per}^{1, 2} (D^{T}) as λ_{m} ↓ λ} . \end{matrix}$ Notice that ${\tilde{G}}_{λ}^{T}$ is a non-empty set for all $λ \in [λ_{1 P}^{T}, + \infty)$ . Indeed, by Lemma B.1, if $λ \in [λ_{1 P}^{T}, + \infty)$ and $λ_{m} ↓ λ$ , then there exist $u_{λ} \in G_{λ}^{T}$ and $u_{λ_{m}} \in G_{λ_{m}}^{T}$ , $m = 1, \dots$ such that $u_{λ_{m}} \to u_{λ}$ strongly in $W_{0, per}^{1, 2} (D^{T})$ as $m \to + \infty$ . We introduce $\begin{matrix} Ξ : = {λ \in [λ_{1 P}^{T}, + \infty) : P_{λ, T} (u_{λ}) < 0, \forall u_{λ} \in {\tilde{G}}_{λ}^{T}} . \end{matrix}$
Proposition 5.2.
Ξ is a non-empty open subset of $(λ_{1 P}^{T}, + \infty)$ .
Proof.
By Lemma 4.2, for any $λ > λ_{0}^{T} > λ_{1 P}^{T}$ there exists $u_{λ} \in M_{λ, T}$ such that $Φ_{λ, T} (u_{λ}) = {\hat{Φ}}_{λ, T}$ . Note that from Lemma 4.1, we have ${\hat{Φ}}_{λ, T} < 0$ . Now, in view of the identity (2.2), we obtain that $P_{λ, T} (u_{λ}) < 0$ , that is $λ \in Ξ$ . Hence $(λ_{0}^{T}, + \infty) \subset Ξ$ and $Ξ \neq \emptyset$ .

Suppose, contrary to our claim, that $λ_{1 P}^{T} \in Ξ$ . Then $Φ_{λ_{1 P}^{T}, T}^{'} (u) = 0$ , $P_{λ_{1 P}^{T}, T} (u) < 0$ , for some $u \in {\tilde{G}}_{λ_{1 P}^{T}}^{T}$ which means that $R_{T}^{P} (u) < λ_{1 P}^{T} = R_{T}^{1} (u)$ . Hence by Lemma 3.2, $1 > t_{1 P} (u)$ . Since ${(R_{T}^{1})}^{'} (t u) > 0$ for $t > t_{1 P} (u) > t_{1} (u)$ , $λ_{1 P}^{T} (u) = R_{T}^{1} (t_{1 P} (u) u) < R_{T}^{1} (u) = λ_{1 P}^{T}$ . However, this contradicts the definition (3.10) of $λ_{1 P}^{T}$ .

Let us prove that Ξ is an open set. On the contrary assume there exist $λ \in Ξ$ and a sequence $(λ_{m}) \subset (λ_{1 P}^{T}, + \infty) ∖ Ξ$ such that $λ_{m} \to λ$ , as $m \to \infty$ . This means that there exists a sequence $u_{λ_{m}} \in {\tilde{G}}_{λ_{m}}^{T}$ such that $P_{λ_{m}, T} (u_{λ_{m}}) = 0$ . Then by Lemma B.2, there exist $u_{λ} \in {\tilde{G}}_{λ}^{T}$ and a subsequence $(u_{λ_{m}})$ , still denoted by $(u_{λ_{m}})$ , such that $u_{λ_{m}} \to u_{λ}$ strongly in $W_{0, per}^{1, 2} (D^{T})$ as $m \to + \infty$ . Hence $u_{λ} \in {\tilde{G}}_{λ}^{T}$ and $P_{λ, T} (u_{λ}) = 0$ , which contradicts $λ \in Ξ$ . □

We introduce $\begin{matrix} λ^{} (T) : = sup {λ \in [λ_{1 P}^{T}, + \infty) : \exists u_{λ} \in {\tilde{G}}_{λ}^{T} with P_{λ, T} (u_{λ}) = 0} . \end{matrix}$ From the above and Lemma B.1 one can conclude that $λ^{} (T) \notin Ξ$ and $(λ^{} (T), + \infty) \subseteq Ξ$ . Lemma 5.1.
For any* $T > 0$ ,
there exists a minimizer $u_{λ^{} (T)}^{T}$ of (* 4.1 ) with compact support in Ω,

$u_{λ^{} (T)}^{T}$ is a least energy solution of (* 1.1 )–( 1.4 ) with $λ = λ^{} (T)$ and* $u_{λ^{} (T)}^{T} ⩾ 0$ in Ω,

$λ_{1 P}^{T} ⩽ λ^{} (T) < λ_{0}^{T}$ .

Proof.
By construction $λ^{} (T) \in [λ_{1 P}^{T}, + \infty)$ , and therefore, ${\tilde{G}}_{λ^{} (T)}^{T} \neq \emptyset$ . Since $λ^{} (T) \notin Ξ$ , there exists $u_{λ^{} (T)}^{T} \in {\tilde{G}}_{λ^{} (T)}^{T}$ such that $P_{λ^{} (T), T} (u_{λ^{} (T)}^{T}) = 0$ . By (5.3), there exists a sequence $λ_{m} \in (λ^{} (T), + \infty) \subseteq Ξ$ , $m = 1, \dots$ such that $λ_{m} ↓ λ^{} (T)$ as $m \to \infty$ , and a sequence $u_{λ_{m}} \in G_{λ_{m}}^{T}$ , $m = 1, 2, \dots$ , such that $u_{λ_{m}} \to u_{λ^{} (T)}^{T}$ strongly in $W_{0, per}^{1, 2} (D^{T})$ as $m \to + \infty$ . Since $λ_{m} \in Ξ$ , $P_{λ_{m}, T} (u_{λ_{m}}) < 0$ , $m = 1, 2, \dots$ , and thus, Proposition 5.1 implies that $u_{λ_{m}}$ , $m = 1, 2, \dots$ are weak nonnegative solutions of (1.1)–(1.4). This and the strong convergence $u_{λ_{m}} \to u_{λ^{} (T)}^{T}$ in $W_{0, per}^{1, 2} (D^{T})$ yield that $u_{λ^{} (T)}^{T}$ is a nonnegative weak solution of (1.1)–(1.4). Then from elliptic regularity theory (see [17]) one gets that $u_{λ^{} (T)}^{T} \in C^{1, γ} (\overline{D^{T}}) \cap C^{2} (D^{T})$ for some $γ \in (0, 1)$ . Hence there holds the Pohozaev identity. Since $P_{λ^{} (T), T} (u_{λ^{} (T)}^{T}) = 0$ , Corollary 2.1 implies that $u_{λ^{} (T)}^{T}$ has a compact support in Ω, and since $u_{λ^{} (T)}^{T}$ is a minimizer of (4.1), $u_{λ^{} (T)}^{T}$ is a least energy solution of (1.1)–(1.4).

To conclude the proof, it is sufficient to show that $λ^{} (T) < λ_{0}^{T}$ . From the proof of Proposition 5.2 we have already $λ^{} (T) ⩽ λ_{0}^{T}$ . Let us assume that $λ^{} (T) = λ_{0}^{T}$ . By Lemma 3.1, there exists ${\hat{u}}_{0}^{T}$ such that $D Φ_{λ^{} (T), T} ({\hat{u}}_{0}^{T}) = 0$ and $Φ_{λ^{} (T), T} ({\hat{u}}_{0}^{T}) = 0$ . Then ${\hat{u}}_{0}^{T} \in M_{λ^{} (T), T}$ and therefore, $Φ_{λ^{} (T), T} (u_{λ^{} (T)}^{T}) ⩽ Φ_{λ^{} (T), T} ({\hat{u}}_{0}^{T}) = 0$ . But $u_{λ^{} (T)}^{T}$ is a solution with compact support, and thus $P_{λ^{} (T), T} (u_{λ^{} (T)}^{T}) = 0$ . This by equality (2.2) implies the opposite inequality $Φ_{λ^{} (T), T} (u_{λ^{} (T)}^{T}) > 0$ . □
Proof of Theorem 1.2.
By Lemma 5.1, we have $λ^{} (T) \in [λ_{1 P}^{T}, λ_{0}^{T})$ . Moreover, from Lemma 4.2, for any $λ ⩾ λ^{} (T)$ , there exists a minimizer $u_{λ}^{T}$ of (4.1) such that $P_{λ, T} (u_{λ}^{T}) ⩽ 0$ . Since $(λ^{} (T), + \infty) \subset Ξ$ , for all $λ > λ^{} (T)$ , we can find a minimizer $u_{λ}^{T}$ of (4.1) such that $P_{λ, T} (u_{λ}^{T}) < 0$ . Then, on account of Proposition 5.1, we deduce that $u_{λ}^{T}$ is a weak solution of (1.1)–(1.4). By Lemma 5.1, there exists a minimizer $u_{λ^{} (T)}^{T}$ which is a least energy solution of (1.1)–(1.4) with $λ = λ^{} (T)$ . To complete the proof of $(1^{o})$ for $λ ⩾ λ^{} (T)$ , we observe that
From Proposition 3.1 (i), for $λ \in [λ^{} (T), λ_{0}^{T})$ , we have $Φ_{λ, T} (u_{λ}^{T}) > 0$ .

For $λ = λ_{0}^{T}$ , on account of Lemma 3.1, we have the existence of a weak solution satisfying $Φ_{λ, T} (u_{λ}^{T}) = 0$ .

From the proof of Proposition 5.2, we have ${\hat{Φ}}_{λ, T} < 0$ , that is, $Φ_{λ, T} (u_{λ}^{T}) < 0$ , for any $λ > λ_{0}^{T}$ .
Lemma 2.2 implies that $Φ_{λ, T}^{″} (u_{λ}^{T}) > 0$ . As above, from elliptic regularity theory we have $u_{λ}^{T} \in C^{1, γ} (\overline{D^{T}}) \cap C^{2} (D^{T})$ for some $γ \in (0, 1)$ . The proof of $(2^{o})$ is clear from parts 1 and 2 of Lemma 5.1. Since $(λ^{} (T), + \infty) \subset Ξ$ , for any $λ \in (λ^{} (T), + \infty)$ there holds $P_{λ, T} (u_{λ}) < 0$ , $\forall u_{λ} \in {\tilde{G}}_{λ}^{T}$ . Hence Corollary 2.1 implies that the periodic least energy solution $u_{λ} \in {\tilde{G}}_{λ}^{T}$ is not compactly supported in Ω. Furthermore, by above if $λ \in [λ_{0}^{T}, + \infty)$ , then any least energy solution $u_{λ} \in G_{λ}^{T}$ satisfies $Φ_{λ, T} (u_{λ}^{T}) ⩽ 0$ and consequently, by (2.2), we have $P_{λ, T} (u_{λ}^{T}) < 0$ . This concludes the proof of $(3^{o})$ . The proof of $(4^{o})$ follows from Corollary 3.2. □

6. Proof of Theorem 1.3

Proof of $(1^{o})$ . Suppose, contrary to our claim, that there is a sequence $(T_{m})$ , $T_{m} \to 0$ as $m \to + \infty$ such that for every $m = 1, 2, \dots$ , there exists a least energy solution $u_{λ^{*} (T_{m})}^{T_{m}}$ of (1.1)–(1.4) which has a compact support in $D_{T_{m}}$ . By Appendix A, problem $(P (D_{T_{m}}))$ has no compact support solution for $λ < λ_{*} (D_{T_{m}})$ . Hence $λ^{*} (T_{m}) ⩾ λ_{*} (D_{T_{m}})$ . This by (A.1) implies that $λ^{*} (T_{m}) \to + \infty$ as $m \to + \infty$ .

Notice that $W_{0}^{1, 2} (Ω)$ can be identified with $W_{c} : = {u \in W_{0, per}^{1, 2} (D^{T}) : \int_{D^{T}} | u_{z} |^{2} d x d z = 0}$ and $\begin{aligned} λ_{0}^{T} (u) & = c_{0}^{q, p} \frac{{(2 T \int_{Ω} | u |^{q + 1} d x)}^{\frac{1 - p}{1 - q}} {(2 T \int_{Ω} | \nabla_{x} u |^{2} d x)}^{\frac{p - q}{1 - q}}}{2 T \int_{Ω} | u |^{p + 1} d x} \\ = c_{0}^{q, p} \frac{{(\int_{Ω} | u |^{q + 1} d x)}^{\frac{1 - p}{1 - q}} {(\int_{Ω} | \nabla_{x} u |^{2} d x)}^{\frac{p - q}{1 - q}}}{\int_{Ω} | u |^{p + 1} d x} = : λ_{0}^{Ω} (I_{Ω} u), \forall u \in W_{c}, \forall T > 0, \end{aligned}$ where $I_{Ω} u (\cdot, x) = u (0, x)$ , $x \in Ω$ for $u \in W_{c}$ , $c_{0}^{q, p}$ is given by (3.3). Hence $\begin{matrix} λ_{0}^{T} = inf_{u \in W_{0, per}^{1, 2} (D^{T}) ∖ {0}} λ_{0}^{T} (u) ⩽ inf_{u \in W_{0}^{1, 2} (Ω) ∖ {0}} λ_{0}^{Ω} (u) = : λ_{0}^{Ω}, \forall T > 0, \end{matrix}$ and thus by Lemma 5.1 we have $λ^{*} (T_{m}) < λ_{0}^{T_{m}} ⩽ λ_{0}^{Ω} < + \infty$ for all $m = 1, 2, \dots$ , which means a contradiction.

Proof of $(2^{o})$ . To prove the claim, suppose on the contrary, that there exists a sequence $(T_{m})$ such that $T_{m} \to + \infty$ as $m \to + \infty$ and for any $m = 1, 2, \dots$ , (1.1)–(1.4) has a periodically trivial compactly supported in Ω least energy solution $u_{λ^{*} (T_{m})}^{T_{m}}$ . From Theorem 1.1 and Remark 1.1, it follows that $u_{λ^{*} (T_{m})}^{T_{m}} (z, \cdot) \equiv ψ_{λ^{*} (T_{m})}^{N} (\cdot)$ , $\forall z \in [- T_{m}, T_{m}]$ , where $ψ_{λ^{*} (T_{m})}^{N} (x) : = {(σ_{λ^{*} (T_{m})})}^{\frac{2}{1 - q}} \cdot ψ^{N} (x / σ_{λ^{*} (T_{m})})$ , $x \in Ω$ , $σ_{λ^{*} (T_{m})} = {(λ^{*} (T_{m}))}^{- \frac{1 - q}{2 (p - q)}}$ . Let us show that $λ^{*} (T_{m}) = λ^{*} (Ω)$ , $m = 1, 2, \dots$ . By Theorem A.1 problem ( $P (Ω)$ ) has no weak solution for $λ < λ^{*} (Ω)$ , and therefore, $λ^{*} (T_{m}) ⩾ λ^{*} (Ω)$ , $\forall m = 1, \dots$ . Suppose, contrary to our claim, that $λ^{*} (T_{m}) > λ^{*} (Ω)$ , for some $m = 1, 2, \dots$ . By Theorem A.1, for any $λ > λ^{*} (Ω)$ problem ( $P (Ω)$ ) possesses least energy solution $u_{λ}^{Ω}$ . Moreover, each least energy solution $u_{λ}^{Ω}$ for $λ \in (λ^{*} (Ω), + \infty)$ is a “usual” solution. In view of that $ψ_{λ^{*} (T_{m})}^{N}$ is a compactly supported solution, we have $Φ_{λ^{*} (T_{m})} (u_{λ}^{Ω}) < Φ_{λ^{*} (T_{m})} (ψ_{λ^{*} (T_{m})}^{N})$ . Set ${\tilde{u}}_{λ}^{Ω} (z, x) = u_{λ}^{Ω} (x)$ for $x \in Ω$ , $z \in [- T_{m}, T_{m}]$ . Then ${\tilde{u}}_{λ}^{Ω}$ satisfies (1.1)–(1.4) and $Φ_{λ^{*} (T_{m}), T_{m}} ({\tilde{u}}_{λ}^{Ω}) < Φ_{λ^{*} (T_{m}), T_{m}} (u_{λ^{*} (T_{m})}^{T_{m}})$ . We get a contradiction since $u_{λ^{*} (T_{m})}^{T_{m}}$ is a least energy solution of (1.1)–(1.4). We thus have $λ^{*} (T_{m}) = λ^{*} (Ω)$ , $\forall m = 1, 2, \dots$ , and therefore, $\begin{matrix} (6.1) & Φ_{λ^{*} (Ω), T_{m}} (u_{λ^{*} (Ω)}^{T_{m}}) = 2 T_{m} Φ_{λ^{*} (Ω)} (ψ_{λ^{*} (Ω)}^{N}) \to + \infty as m \to + \infty . \end{matrix}$ Proposition A.1 yields that there exist $T_{λ^{*} (Ω)} > 0$ and $ϕ_{λ^{*} (Ω)} \in C_{0}^{\infty} (R \times Ω)$ such that $π^{T_{m}} (ϕ_{λ^{*} (Ω)}) = ϕ_{λ^{*} (Ω)} \in M_{λ^{*} (Ω), T_{m}}$ for any $T_{m} > T_{λ^{*} (Ω)}$ . This implies that $\begin{matrix} {\hat{Φ}}_{λ^{*} (Ω), T_{m}} ⩽ Φ_{λ^{*} (Ω), T_{m}} (ϕ_{λ^{*} (Ω)}) < + \infty, \forall T_{m} > T_{λ^{*} (Ω)}, \end{matrix}$ which contradicts (6.1).

7. Conclusions and open problems

We proved for the equation with non-Lipschitz non-linearity the existence of least energy solutions periodic in one variable and subject to the zero Dirichlet conditions on $\partial Ω$ for other variables. Moreover, we find an upper threshold $λ_{1 P}^{T}$ for the nonexistence of solutions of the problem. We believe that the point $λ^{*} (T)$ in Theorem 1.2 is a limit value for the existence of nonnegative solutions of the problem and it corresponds to the turning point bifurcation of a branch of the nonnegative solutions. Note that the main difficulty that has been overcome in this result is obtaining solutions with positive energy $Φ_{λ, T} (u_{λ}^{T}) > 0$ for $λ \in [λ^{*} (T), λ_{0}^{T})$ . Apparently, the least energy solutions for $λ \in (λ_{0}^{T}, + \infty)$ can be obtained without assumption $N (1 - q) (1 - p) - 2 (1 + q) (1 + p) > 0$ , by direct application of the Nehari manifold method. Furthermore, we expect that if $λ \in [λ_{0}^{T}, + \infty)$ , the second branch of nonnegative solutions (1.1)–(1.4) can be obtained by the mountain pass theorem. However, we do not know whether it is possible to construct the second branch of solutions for $λ \in (λ^{*} (T), λ_{0}^{T})$ and whether it forms with the branch of nonnegative solutions $u_{λ}^{T}$ a turning point bifurcation at the value $λ^{*} (T)$ .

Our second result addresses the possibility of the existence of compactly supported solutions of (1.1)–(1.4) with respect to part of the variables $x \in R^{N}$ . Theorems 1.2, 1.3 confirm a positive answer. However, it does not give a complete answer as to whether they are new type solutions with compact supports for equation (1.1). In fact, we conjecture that there exist $T_{1}^{*}$ , $T_{2}^{*}$ , $0 < T_{1}^{*} < T_{2}^{*} < + \infty$ such that

for each $T \in (0, T_{1}^{*})$ , any compactly supported least energy solution $u_{λ^{*} (T)}^{T}$ of (1.1)–(1.4) is periodically trivial, i.e. $\int_{D^{T}} | {(u_{λ^{*} (T)})}_{z} |^{2} d x d z = 0$ .

for any $T \in (T_{1}^{*}, T_{2}^{*})$ the compactly supported least energy solution $u_{λ^{*} (T)}^{T}$ of (1.1)–(1.4) is periodically non-trivial and has no compact support in $D^{T}$ .

for any $T ⩾ T_{2}^{*}$ , any compactly supported periodic least energy solution $u_{λ^{*} (T)}^{T}$ of (1.1)–(1.4) is, in fact, a compactly supported solution of $(1_{R^{N + 1}}^{*})$ .

An interesting question that also remains open is whether compactly supported solutions $u_{λ^{*} (T)}^{T} (z, x)$ are radially symmetric with respect to variables $x \in R^{N}$ , as is the case of problem ( P λ ( Θ ) ) by Theorem A.1 (see also [8,28,38]).

It is worth noting that the results obtained in this article give reason to expect that problem (1.1)–(1.4) can have periodically non-trivial nonnegative least energy solutions, which partially satisfy the Hopf maximum principle on the boundary $\partial Ω$ . A similar result was recently obtained in [5] for another problem.

Footnotes

Acknowledgements

The first author was funded by IFCAM (Indo-French Centre for Applied Mathematics) IRL CNRS 3494.

Further investigations of Theorem 1.1

Let $M ⩾ 2$ . We denote by $R_{M}$ the radius of the supporting ball $B_{R_{M}}$ of the unique (up to translation in $R^{M}$ ) compactly supported solution $ψ^{M}$ of ( 1 R M ∗ ).

Let Θ be a domain in $R^{M}$ . The largest ball $B_{R^{*} (Θ)}^{a} : = {x \in R^{M} : | x - a | ⩽ R^{*} (Θ)}$ with some $a \in Θ$ contained in Θ is said to be inscribed ball in Θ. In what follows, we always assume that the centre a of such ball coincides with $0 \in R^{M}$ and will use the notation $B_{R^{*} (Θ)} : = B_{R^{*} (Θ)}^{0}$ . Thus, $\begin{matrix} R^{*} (Θ) = max {R : B_{R} \subseteq Θ} . \end{matrix}$ Let $R \in (0, R^{*} (Θ))$ and $σ_{R} = R / R_{M}$ . Then the function $ψ_{λ_{R}}^{M} (y) : = {(σ_{R})}^{\frac{2}{1 - q}} \cdot ψ^{M} (y / σ_{R})$ is a compactly supported with $supp (ψ_{λ_{R}}^{M}) = B_{R} \subset Θ$ solution of $\begin{matrix} P_{λ} (Θ) & \{\begin{array}{ll} - Δ u = λ | u |^{p - 1} u - | u |^{q - 1} u & in Θ, \\ u = 0 & on \partial Θ, \end{array} \end{matrix}$ where $\begin{matrix} λ = λ_{R} : = {(σ_{R})}^{- \frac{2 (p - q)}{1 - q}} = {(\frac{R_{M}}{R})}^{\frac{2 (p - q)}{1 - q}} . \end{matrix}$ In what follows, we denote $\begin{matrix} λ^{*} (Θ) : = {(\frac{R_{M}}{R^{*} (Θ)})}^{\frac{2 (p - q)}{1 - q}} and ψ_{λ^{*} (Θ)}^{M} : = ψ_{λ_{R^{*} (Θ)}}^{M} . \end{matrix}$ Note that $\begin{aligned} (A.1) & λ_{R} \to + \infty as R \to 0 . \end{aligned}$ Define $\begin{matrix} Λ_{1 P}^{Θ} = inf_{u \in W_{0}^{1, 2} (Θ) ∖ {0}} Λ_{1 P}^{Θ} (u), \end{matrix}$ where $\begin{matrix} Λ_{1 P}^{Θ} (u) = c_{1 P}^{q, p, 0} \frac{{(\int_{Θ} | u |^{q + 1} d x)}^{\frac{1 - p}{1 - q}} {(\int_{Θ} | \nabla u |^{2} d x)}^{\frac{p - q}{1 - q}}}{\int_{Θ} | u |^{p + 1} d x} \end{matrix}$ and $\begin{matrix} c_{1 P}^{q, p, 0} = \frac{(p + 1) (2_{M}^{*} - q - 1)}{{(2_{M}^{*} (p - q))}^{\frac{p - q}{1 - q}}} {(\frac{1}{(2_{M}^{*} - p - 1) (q + 1)})}^{\frac{1 - p}{1 - q}} . \end{matrix}$ By [15, Corollary 3.1], if $λ < Λ_{1 P}^{Θ}$ and $Φ_{λ, Θ}^{'} (u) = 0$ , then $P_{λ, Θ} (u) > 0$ , while for any $λ > Λ_{1 P}^{Θ}$ , there exists $u \in W_{0}^{1, 2} (Θ) ∖ {0}$ such that $Φ_{λ, Θ}^{'} (u) = 0$ and $P_{λ, Θ} (u) < 0$ . Here $\begin{matrix} P_{λ, Θ} (u) : = \frac{1}{2_{M}^{*}} \int_{Θ} | \nabla u |^{2} d x d z + \frac{1}{q + 1} \int_{Θ} | u |^{q + 1} d x d z - \frac{λ}{p + 1} \int_{Θ} | u |^{p + 1} d x d z . \end{matrix}$ In what follows if $u \in C^{1} (\overline{Θ}) \cap C^{2} (Θ) ∖ {0}$ is a solution of ( P λ ( Θ ) ) such that $\frac{\partial u (z)}{\partial ν} \neq 0$ for some $x \in \partial Θ$ we shall call it as a “usual” solution of ( P λ ( Θ ) ).

The following result follows from [15] (see Sections 4 and 5).

Define $π^{T} (u) (z, \cdot) = u (z, \cdot)$ , $z \in [- T, T]$ , $u \in C^{\infty} (R \times Ω)$ . Proposition A.1.

For any $λ ⩾ λ^{*} (Ω)$ , there exist $T_{λ} > 0$ and $ϕ_{λ} \in C_{0}^{\infty} (R \times Ω)$ such that $π^{T} (ϕ_{λ}) \in M_{λ, T}$ for any $T > T_{λ}$ .

Proof.

By Theorem A.1, one has $Λ_{1 P}^{Ω} < λ^{*} (Ω)$ . Let us prove that $λ_{1 P}^{\infty} ⩽ Λ_{1 P}^{Ω}$ . For that consider u a minimizer of $λ_{1 P}^{Ω}$ and for any $T > 0$ the even cut-off function $ϕ_{T}$ defined in $R^{+}$ as $ϕ_{T} (t) = 1$ if $t \in [0, T]$ , $ϕ_{T} (t) = (T + 1 - t)$ if $t \in [T, T + 1]$ and $ϕ_{T} (t) = 0$ if $t ⩾ T + 1$ . Then, $\begin{array}{rcl} λ_{1 P}^{\infty} & ⩽ & λ_{1 P}^{\infty} (ϕ_{T} u) \\ = & c_{1 P}^{q, p} \frac{((2 T + \frac{2}{3}) (\frac{2_{N}^{*} - 1 - q}{2_{N}^{*}}) \int_{Ω} | \nabla_{x} u |^{2} d x + \frac{1 - q}{2} \int_{Ω} u^{2} d x) {((2 T + \frac{2}{q + 2}) \int_{Ω} | u |^{q + 1})}^{\frac{1 - p}{1 - q}}}{(2 T + \frac{2}{p + 2}) \int_{Ω} | u |^{p + 1} d x) {((2 T + \frac{2}{3}) \frac{2_{N}^{*} - 1 - p}{2_{N}^{*}} \int_{Ω} | \nabla_{x} u |^{2} d x + \frac{1 - q}{2} \int_{Ω} u^{2} d x)}^{\frac{1 - p}{1 - q}}} . \end{array}$ Thus, as $T \to \infty$ , $λ_{1 P}^{\infty} (ϕ_{T} u) \sim Λ_{1 P}^{Ω} (u)$ . Passing the limit as $T \to \infty$ in the above expression, we obtain $λ_{1 P}^{\infty} ⩽ Λ_{1 P}^{Ω} < λ$ . Therefore, there exists $w \in W_{0, per}^{1, 2} (D^{\infty}) ∖ {0}$ such that $λ_{1 P}^{\infty} < λ_{1 P}^{\infty} (w) < λ$ . In view of that $C_{0}^{\infty} (R \times Ω)$ is dense in $W_{0, per}^{1, 2} (D^{\infty})$ and $λ_{1 P}^{T} (\cdot) \in C^{1} (W_{0}^{1, 2} (D^{T}) ∖ {0})$ , we obtain $ϕ \in C_{0}^{\infty} (R \times Ω) ∖ {0}$ such that $λ_{1 P}^{\infty} < λ_{1 P}^{\infty} (ϕ) < λ$ . This implies $P_{λ, \infty} (ϕ) < 0$ , $Φ_{λ, \infty}^{'} (ϕ) < 0$ , and therefore, there exists $t_{λ} (ϕ) > 1$ such that $Φ_{λ, \infty}^{'} (t_{λ} (ϕ) ϕ) = 0$ . By Proposition 2.1, we have $\frac{d P_{λ, \infty} (t ϕ)}{d t} < 0$ for $t \in [1, t_{λ} (ϕ)]$ . Thus $P_{λ, \infty} (t_{λ} (ϕ) ϕ) < 0$ and $Φ_{λ, \infty}^{'} (t_{λ} (ϕ) ϕ) = 0$ . Since ϕ has a compact support in $D^{\infty} = R \times Ω$ , there exists $T_{λ} > 0$ such that $\forall T > T_{λ}$ , $ϕ (- T, \cdot) = ϕ (T, \cdot) = 0$ , and $\begin{aligned} P_{λ, T} (t_{λ} (ϕ) ϕ) = P_{λ, \infty} (t_{λ} (ϕ) ϕ) < 0, \\ Φ_{λ, T}^{'} (t_{λ} (ϕ) ϕ) = Φ_{λ, \infty}^{'} (t_{λ} (ϕ) ϕ) = 0, \end{aligned}$ which means that $ϕ_{λ} : = t_{λ} (ϕ) ϕ$ satisfies $π^{T} (ϕ_{λ}) \in M_{λ, T}$ for any $T > T_{λ}$ . □

A technical result about minimizers

A Pohozaev type inequality

In this section, we show Lemma 2.1. We prove it in the general setting: $\begin{matrix} \{\begin{array}{ll} - Δ u = g (u), & (z, x) \in (- T, T) \times Ω, \\ u (- T, x) = u (- T, x), u_{z} (- T, x) = u_{z} (- T, x), & x \in Ω, \\ u (z, x) = 0, & x \in \partial Ω, z \in (- T, T) . \end{array} \end{matrix}$

References

G.L.

Alfimov ,

V.M.

Eleonsky ,

N.E.

Kulagin ,

L.M.

Lerman and

V.P.

Silin , On existence of nontrivial solutions for the equation

Δ u - u + u^{3} = 0

, Physica D: Nonlinear Phenomena 44(1–2) (1990), 168–177. doi:10.1016/0167-2789(90)90053-R.

S.N.

Antontsev ,

J.I.

Díaz ,

Shmarev and

A.J.

Kassab , Energy methods for free boundary problems: Applications to nonlinear PDEs and fluid mechanics. Progress in nonlinear differential equations and their applications, Vol 48, Appl. Mech. Rev. 55(4) (2002), B74–B75. doi:10.1115/1.1483358.

Berestycki and

P.-L.

Lions , Nonlinear scalar field equations, I existence of a ground state, Archive for Rational Mechanics and Analysis 82(4) (1983), 313–345. doi:10.1007/BF00250555.

Berestycki and

Wei , On least energy solutions to a semilinear elliptic equation in a strip, Discrete and Continuous Dynamical Systems 28(3) (2010), 1083–1099. doi:10.3934/dcds.2010.28.1083.

Bobkov ,

Drábek and

Ilyasov , On partially free boundary solutions for elliptic problems with non-Lipschitz nonlinearities, Applied Mathematics Letters 95 (2019), 23–28. doi:10.1016/j.aml.2019.03.019.

Busca and

Felmer , Qualitative properties of some bounded positive solutions to scalar field equations, Calc. Var. Partial Differential Equations 13(2) (2001), 191–211. doi:10.1007/PL00009928.

Cortázar ,

Elgueta and

Felmer , Symmetry in an elliptic problem and the blow-up set of a quasilinear heat equation, Comm. Partial Differential Equations 21 (1996), 507–520. doi:10.1080/03605309608821194.

Cortázar ,

Elgueta and

Felmer , On a semi-linear elliptic problem in

R^{N}

with a non-Lipschitzian non-linearity, Advances in Diff. Equs. 1 (1996), 199–218.

Dancer , New solutions of equations on

R^{n}

, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 30(4) (2001), 535–563.

10.

Del Pino ,

Kowalczyk ,

Pacard and

Wei , The Toda system and multiple-end solutions of autonomous planar elliptic problems, Advances in Mathematics 224(4) (2010), 1462–1516. doi:10.1016/j.aim.2010.01.003.

11.

Delaunay , Sur les surfaces de révolution dont la courbure moyenne est constante (French), Jounal de mathématiques 26 (1898), 43–52.

12.

J.I.

Díaz , Nonlinear partial differential equations and free boundaries. Elliptic equations, Research Notes in Math. 1 (1985), 106.

13.

J.I.

Díaz ,

Hernández and

Il’yasov , On the existence of positive solutions and solutions with compact support for a spectral nonlinear elliptic problem with strong absorption, Nonl. Anal.: Th., Meth. & Appl. 119 (2015), 484–500. doi:10.1016/j.na.2014.11.019.

14.

J.I.

Díaz ,

Hernández and

Il’yasov , Flat and compactly supported solutions of some non-Lipschitz autonomous semilinear equations may be stable for

N ⩾ 3

, Chin. Ann. Math. Ser. B 38(1) (2017), 345–378. doi:10.1007/s11401-016-1073-2.

15.

J.I.

Díaz ,

Hernández and

Y.S.

Ilyasov , On the exact multiplicity of stable ground states of non-Lipschitz semilinear elliptic equations for some classes of starshaped sets, Adv. Nonlinear Anal. 9(1) (2020), 1046–1065. doi:10.1515/anona-2020-0030.

16.

A.I.

Dyachenko ,

A.N.

Pushkarev ,

A.M.

Rubenchik ,

R.Z.

Sagdeev ,

V.F.

Shvets and

V.E.

Zakharov , Computer simulation of Langmuir collapse, Physica D: Nonlinear Phenomena 52(1) (1991), 78–102. doi:10.1016/0167-2789(91)90029-9.

17.

Gilbarg and

N.S.

Trudinger , Elliptic Partial Differential Equations of Second Oder, 2nd edn, Grundlehren, Vol. 224, Springer, Berlin–Heidelberg–New York–Tokyo, 1983.

18.

M.D.

Groves ,

Haragus and

S.M.

Sun , A dimension-breaking phenomenon in the theory of steady gravity-capillary water waves, Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 360(1799) (2002), 2189–2243. doi:10.1098/rsta.2002.1066.

19.

Hernández ,

Mancebo and

J.M.

Vega , Positive solutions for singular nonlinear elliptic equations, Proc. Roy. Soc. Edinburgh Sect. A 137(1) (2007), 41–62. doi:10.1017/S030821050500065X.

20.

Ilyasov , On extreme values of Nehari manifold method via nonlinear Rayleigh’s quotient, Topological Methods in Nonlinear Analysis 49(2) (2017), 683–714.

21.

Ilyasov and

Egorov , Hopf maximum principle violation for elliptic equations with non-Lipschitz nonlinearity, Nonlin. Anal. 72 (2010), 3346–3355. doi:10.1016/j.na.2009.12.015.

22.

Il’yasov and

Takac , Optimal-regularity, Pohozhaev’s identity, and nonexistence of weak solutions to some quasilinear elliptic equations, J. Differential Equations 252(3) (2012), 2792–2822. doi:10.1016/j.jde.2011.10.020.

23.

Y.S.

Il’yasov , On the existence of periodic solutions of semilinear elliptic equations, Mat. Sb. 184(6) (1993), 67–82, Russian Acad. Sci. Sb. Math. 79(1) (1994), 167–178.

24.

Y.S.

Il’yasov , An approximation to ground state solutions for the equation

- Δ u = g (u)

R^{N + 1}

, Differential Equations 30(4) (1994), 570–579.

25.

Y.S.

Il’yasov , On periodic non-trivial solutions of the equation

- Δ u = g (u)

R^{N + 1}

, Izvestiya: Mathematics 59(1) (1995), 101–119. doi:10.1070/IM1995v059n01ABEH000004.

26.

Y.S.

Il’yasov , On the curve of critical exponents for nonlinear elliptic problems in the case of a zero mass, Comput. Math. Math. Phys. 57(3) (2017), 497–514. doi:10.1134/S096554251703006X.

27.

Kaper and

Kwong , Free boundary problems for Emden-Fowler equation, Differential and Integral Equations 3 (1990), 353–362.

28.

Kaper ,

Kwong and

Li , Symmetry results for reaction-diffusion equations, Differential and Integral Equations 6 (1993), 1045–1056.

29.

J.L.

Lions , Quelques méthodes de résolution des problemes aux limites non linéaires, Dunod, Paris, 1969.

30.

Maagli ,

Giacomoni and

Sauvy , Existence of compact support solutions for a quasilinear and singular problem, Differential Integral Equations 25(7–8) (2012), 629–656.

31.

Malchiodi , Some new entire solutions of semilinear elliptic equations on

R^{n}

, Advances in Mathematics 221(6) (2009), 1843–1909. doi:10.1016/j.aim.2009.03.012.

32.

Mazzeo and

Pacard , Constant mean curvature surfaces with Delaunay ends, Comm. Anal. Geom. 9 (2001), 169–237. doi:10.4310/CAG.2001.v9.n1.a6.

33.

P.A.

Milewski and

Wang , Transversally periodic solitary gravity–capillary waves, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 470(2161) (2014), 20130537.

34.

S.I.

Pohozaev , Eigenfunctions of the equation

Δ u + λ f (u) = 0

, Sov. Math. Doklady 5 (1965), 1408–1411.

35.

Pucci and

Serrin , The Maximum Principle, Progress in Nonlinear Differential Equations and Their Applications, Vol. 73, Birkhäuser Verlag, Basel, 2007.

36.

P.A.

Robinson , Nonlinear wave collapse and strong turbulence, Reviews of modern physics 69(2) (1997), 507–573. doi:10.1103/RevModPhys.69.507.

37.

P.A.

Robinson ,

D.L.

Newman and

M.V.

Goldman , Three-dimensional strong Langmuir turbulence and wave collapse, Physical Review Letters 61(6) (1988), 702. doi:10.1103/PhysRevLett.61.702.

38.

Serrin and

Zou , Symmetry of ground states of quasilinear elliptic equations, Archive for Rational Mechanics and Analysis 148(4) (1999), 265–290. doi:10.1007/s002050050162.

39.

J.L.

Vázquez , A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim. 12 (1984), 191–202. doi:10.1007/BF01449041.

40.

V.E.

Zakharov , Stability of periodic waves of finite amplitude on the surface of a deep fluid, Journal of Applied Mechanics and Technical Physics 9(2) (1968), 190–194. doi:10.1007/BF00913182.

41.

Zhan and

P.A.

Milewski , Dynamics of gravity–capillary solitary waves in deep water, Journal of fluid mechanics 708 (2012), 480–501. doi:10.1017/jfm.2012.320.

42.

A.A.

Zozulya ,

D.Z.

Anderson ,

A.V.

Mamaev and

Saffman , Solitary attractors and low-order filamentation in anisotropic self-focusing media, Physical Review A 57(1) (1998), 522–534. doi:10.1103/PhysRevA.57.522.