Application of a Variant of Mountain Pass Theorem in Modeling Real Phenomena

Abstract

Mountain Pass Theorem (MPT) is an important result in variational methods with multiple applications in partial differential equations involved in mathematical physics. Starting from a variant of MPT, a new result concerning the existence of the solution for certain mathematical physics problems involving p-Laplacian and p-pseudo-Laplacian has been obtained. Based on the main theorem, the existence, possibly the uniqueness, and characterization of solutions for models such as nonlinear elastic membrane, glacier sliding, and pseudo torsion problem have been obtained. The novelty of the work consists of the formulation of the central result under weaker conditions requested by the chosen variant of MPT, the proof of this statement, and its application in solving above mentioned problems. While the expressions of such Dirichlet and/or von Neumann problems were already completed, this proposed solving method suggests some specific numerical methods to construct the appropriate solution. A general goal of this paper is the extension of the applicative pallet of this way to construct the solutions encountered in modeling real processes developed within new emerging technologies.

1. Introduction

The Mountain Pass Theorem (here and in many other papers abbreviated in MPT), reported by Ambrosetti and Rabinowitz (1973), is a famous result in nonlinear analysis, with numerous applications, especially in partial differential equations.

When taking into account their importance in applications, p-Laplacian and p-pseudo-Laplacian have attracted special attention. They appear in mathematical models for subjects such as glaciology [1,2,3], nonlinear diffusion [4,5,6,7], filtration problems [8], power-law materials [9,10], non-Newtonian fluids [1,11], reaction–diffusion problems [12], flow through porous media [13], nonlinear elasticity [14,15], petroleum extraction [16,17], torsional creep problems [14,15], etc. For physical background and interpretations, the paper [18] may be consulted. The nonlinear boundary condition describes a flux through the boundary ∂Ω, which depends on the solution itself. For a physical interpretation of such conditions, the paper [9] is relevant.

There is an increasing interest for these kinds of operators both for mathematical approaches and a series of other results such as in [4,19,20,21,22,23,24,25,26,27,28], as well as for applications in many models in various fields such as in [7,28,29,30,31,32].

Previously, the author reported on similar problems of mathematical physics using variational methods, e.g., via the Ekeland variational principle or some generalized variational principles in [33,34,35,36,37] and by the usage of subjectivity methods such as in [36,37,38,39]. The contributions of the author on variants of the minimax theorem and, consequently, on the mountain pass theorem can be found in [40] and, more extended, in [41].

In this paper, starting from the variant of the mountain pass theorem formulated in [40], an existence result on weak solutions for some mathematical physics problems involving p-Laplacian and p-pseudo-Laplacian has been proven. Furthermore, some applications of such methods in obtaining and characterization of solutions for special models from glaciology, deformation of a nonlinear elastic membrane, and pseudo torsion problems issued from mode-ling real phenomena have been carried out. This way to find out the solution using results of MPT type directs towards some specific numerical methods and hence to the special construction of the solution.

The novelty of this work resides in the application of the mentioned MPT variant in obtaining the main result (which is also a generalization of a similar result from [42]), the proof of this central theorem, and its use in solving some existing problems issued from real phenomena modeling. This paper contains a sequence of results, starting from the abstract frame of the considered variant of MPT, passing step by step through many concrete stages until their applications in solving some real models. Even though such models have been proposed in different papers with the solutions obtained via other methods, we provide here original methods following our abstract frame. A general goal of this paper is the extension of the applicative pallet in this way to construct the solutions representing real modeling processes developed in new emerging technologies.

2. Main Result and Theoretical Background with Comments

Firstly, let us present a generalization from [40] of the MPT variant of Ambrosetti and Rabinowitz given for the goal function of Fréchet C¹-class having the weaker condition φ′_w ( ∙ )(v) upper semicontinuous ∀v, the derivative being of Gâteaux type.

Mountain Pass Theorem.

Let X be a Banach space, φ: X → R continuous and Gâteaux differentiable with φ′_w( ∙ )(v) upper semicontinuous∀v ∈ X. Suppose that for u_0, u₁ from X and Ω an open neighbourhood of u₀ such that u₁ ∈ X\$\overline{\mathsf{\Omega }}$we have

$$\underset{\mathrm{Fr}\mathsf{\Omega }}{\inf } \mathrm{\phi } > \max (\mathrm{\phi }({\mathrm{u}}_0),\mathrm{\phi }({\mathrm{u}}_1)).$$

Let be

Γ: = {g ∈ C([0,1]; X): g(0) = u₀, g(1) = u₁}

and

$$\mathrm{c}:=\underset{g\in \mathsf{\Gamma }}{\inf }\underset{\left[0,1\right]}{ \sup } (\mathrm{\phi } \mathrm{o} \mathrm{g}).$$

If φ verifies the (PS)_c condition, then c is a critical value of φ and c > max (φ(u₀), φ(u₁)) [40].

Clarification 1. Let X be a real normed space. A nonempty set β of bounded parts of X, with the properties: 1^o ${\cup }_{A\in \beta }A$= X; 2^o A ∈ β ⇒ − A ∈ β and λ A ∈ β (λ > 0); 3^o for every A, B in β there exists C in β such that A ⊂ C and B ⊂ C, is named bornology on X.

Let β be a bornology on X and f: X → $\overline{\mathrm{R}}$ locally finite in the point a (there is a neighborhood of a on which f is finite). By definition, f is β-differentiable in a, if there exists φ in the dual X * such that for every S in β we have $\underset{\begin{array}{c}t\to 0\\ h\in S\end{array}}{\mathrm{limu}}\frac{f\left(a+th\right)-f\left(a\right)}{t}$ = φ(h) (uniform limit on S for t → 0). φ is the β-derivative of f in a, and is denoted ∇_β f (a). f β-differentiable on a subset of X has the usual meaning.

Clarification 2. φ verifies the Palais–Smale condition on the level c, (PS)_c, with respect to the bornology β when, ∀(u_n)_n≥1 sequence of points in X for which

$$\underset{n\to \infty }{\lim }\mathrm{\phi }\left(u_n\right)=c \mathrm{and} \underset{n\to \infty }{\lim }\left|\left|{\mathit{\nabla }}_{\beta }\mathsf{\phi }\left(u_n\right)\right|\right|=0,$$

this sequence has a convergent subsequence.

c is a critical value for φ, that is c = φ(u₀) and ∇_βφ(u₀) = 0.

Remark 1.

The intuitive name “Mountain Pass” given to this theorem comes from a topographical situation that takes its source in the case X =R²with the representation in the Euclidean space E₃ of the elementary geometry. The conditions for u₀, u₁ and φ show that the points M₀: = (u_0, φ(u₀)) and M₁: = (u_1, φ(u₁)) are lower then the mountain wall:

{(x, y, φ(x, y)): (x, y) ∈ Fr Ω}

and are situated on both sides of it. The path

t → φ(g(t)), t ∈ [0,1]

starts from M₀ and arrives at M₁ passing the mountain wall, since, as we have seen above,

g([0,1]) ∩ Fr Ω ≠ ∅.

But the highest point of the path has the quota

$$\underset{t\in \left[0,1\right]}{\sup }\mathsf{\phi }\left(g\left(t\right)\right),$$

an attained supremum. Thus, taking into account the significance that c receives now, the ”Mountain pass” intuitive denomination is seen to be well chosen.

Remark 2.

This theorem was obtained by Ambrosetti and Rabinowitz under the assumption φ of Fréchet C¹ - class. The passage to φ continuous with φ′_w ( ∙ )(v) u.s.c.,∀v ∈ X was performed by the author [40]. Obviously, the Gâteaux derivative in the above statement can be replaced by any β-derivative, β bornology on X ([41], Ch. I, 3.10).

Consider the problems:

$$(*) \left\{\begin{array}{l}-{\mathsf{\Delta }}_{p}u\left(x\right)=f\left(x,u\left(x\right)\right) \mathrm{on} \mathsf{\Omega }\\ u\left(x\right)=0 \mathrm{on} \partial \mathsf{\Omega }\end{array}\right.,$$

and

$$(**) \left\{\begin{array}{l}-{\mathsf{\Delta }}_p^{s}u\left(x\right)=f\left(x,u\left(x\right)\right) \mathrm{on} \mathsf{\Omega }\\ u\left(x\right)=0 \mathrm{on} \partial \mathsf{\Omega }\end{array}\right.,$$

where Ω is a bounded smooth domain of R^N. The function f: $\overline{\mathsf{\Omega }}$ × R → R, is supposed to be a Carathéodory function satisfying the growth condition:

|f(x, s)| ≤ c |s|^p−1 + α(x), ∀x ∈ Ω\A with μ(A) = 0 ∀s ∈ R,

where c ≥ 0, α ∈ L^p’(Ω), with $\frac{1}{p}+\frac{1}{p\prime }=1, 1 \le p \le \frac{2N}{N-2}$ if N ≥ 3 and 1 ≤ p ≤ +∞ if N = 2. By definition, f: Ω × R → R is a Carathéodory function if: 1⁰ f (·, s) is Lebesgue measurable ∀s in R; 2⁰ f (x, · ) is continuous ∀x in Ω\A, μ(A) = 0. In this case, for any u from M (Ω), x →f (x, u(x)) is Lebesgue measurable [43].

So, the energy functional φ associated to (∗) is φ: $W_0^{1,p}$(Ω) → R, defined by:

$$\mathsf{\phi }(u)=\frac{1}{p}|\left|u\right|{|}_{1,p}^p -{\int }_{\mathsf{\Omega }}F\left(x,u\left(x\right)\right)dx,$$

and for (∗∗) is φ: $W_0^{1,p}$(Ω) → R, defined by:

$$\mathsf{\phi }(u)=\frac{1}{p}{▍u▍}_{1,p}^p-{\int }_{\mathsf{\Omega }}F\left(x,u\left(x\right)\right)dx,$$

where $F\left(x,s\right)={\int }_0^{s}f\left(x,t\right)dt$, ‖u‖_1,p $=$${\Vert u\Vert }_{L^p\left(\mathsf{\Omega }\right)}+{\sum }_{i=1}^N{\Vert \frac{\partial u}{\partial x_i}\Vert }_{L^p\left(\mathsf{\Omega }\right)}\mathrm{is} \mathrm{a} \mathrm{norm} \mathrm{on} W_0^{1,p}$(Ω) which is equivalent to the norm u →${\left(|\left|u\right|{|}_{L^p\left(\mathsf{\Omega }\right)}^p+{\sum }_{i=1}^N{\Vert \frac{\partial u}{\partial x_i}\Vert }_{L^p\left(\mathsf{\Omega }\right)}^p\right)}^{\frac{1}{p}}$ and it is also equivalent to the norm u → ${\Vert \left|\nabla u\right|\Vert }_{L^p\left(\mathsf{\Omega }\right)}$, and, for the problem (∗∗), ▍u▍_1,: = ${\left({\sum }_{i=1}^N{\Vert \frac{\partial u}{\partial x_i}\Vert }_{L^p\left(\mathsf{\Omega }\right)}^p\right)}^{\frac{1}{p}}$ is another norm on $W_0^{1,p}\left(\Omega \right) \mathrm{equivalent} \mathrm{to} \mathrm{the} \mathrm{norm}: {\left|u\right|}_{1,p}:={\sum }_{i=1}^N{\Vert \frac{\partial u}{\partial x_i}\Vert }_{L^p\left(\mathsf{\Omega }\right)}, \mathrm{where} \Vert u{\Vert }_{L^p\left(\mathsf{\Omega }\right)}:={\left({\int }_{\mathsf{\Omega }}|u{|}^{p}dx\right)}^{\frac{1}{p}}$. The vector space $W_0^{1,p} \left(\Omega \right),$ endowed with one of the first three equivalent norms is a smooth Banach space, and the Banach space ($W_0^{1,p} \left(\Omega \right)$, ▍ · ▍_1,p) is also smooth.

Hence, for (∗),

$$\mathsf{\phi }(u)={\int }_{\mathsf{\Omega }}\left[\frac{1}{p}\left(|u{|}^p+{\sum }_{i=1}^n{\left|\frac{\partial u}{\partial x_i}\right|}^p\right)-F\left(x,u\left(x\right)\right)\right]dx$$

and, for (∗∗),

$$\mathsf{\phi }(u)={\int }_{\mathsf{\Omega }}\left[\frac{1}{p}{\sum }_{i=1}^n{\left|\frac{\partial u}{\partial x_i}\right|}^p-F\left(x,u\left(x\right)\right)\right]dx.$$

Applications based on the mountain pass theorem require conditions on the behavior of f at 0 and for |s| → +∞.

Theorem 1.

Assume that f satisfies:

(f1) 

f (x, s) = o(|s|), s → 0, uniformly for x ∈ $\overline{\Omega }$;

(f2) 

There exist constants μ> p and r > 0 e.g., for|s| ≥ r,

             0 < μF(x, s) ≤ sf (x, s).

Then the problems (∗) and (∗∗) possess a nontrivial solution.

Remark 3.

f (x, s) = o(|s|) for s → 0 uniformly on $\overline{\Omega }$ means

$$\underset{\begin{array}{c}s\to 0\\ x\in \overline{\Omega }\end{array}}{\lim }\mathrm{u}\frac{f\left(x,s\right)}{\left|s\right|} =0 \Rightarrow \forall \mathsf{\epsilon } > 0 \exists \mathsf{\delta } > 0 \mathrm{e}.\mathrm{g}., \left|s\right| \le \mathsf{\delta } \Rightarrow \left|\frac{f\left(x,s\right)}{\left|s\right|}-0\right|\le \mathsf{\epsilon }, \forall x\in \overline{\mathsf{\Omega }}.$$

Integrate on [0, t], |t| ≤ δ: − ε|s| < f (x, s) ≤ ε|s|, for x fixed in $\overline{\mathsf{\Omega }}. 0 \le F\left(x, t\right) \le \frac{\epsilon }{2}$|t|² + C.

By (f1), u ≡ 0 is a trivial solution for (∗) and (∗∗).

Remark 4.

F(x, t) =${\int }_0^{t}f\left(x,s\right)ds and \left|F\left(x, t\right)\right|=\left|{\int }_0^{t}f\left(x,s\right)ds\right|\le {\int }_0^t\left|f\left(x,s\right)\right|ds\le {\int }_0^t[c|s{|}^{p-1}+\alpha \left(x\right)]ds=\frac{c}{p}$|t|^p + t α(x), ∀x ∈ Ω\A with μ(A) = 0 ∀t ∈R.

Remark 5.

For |s| ≥ r, 0 < μF(x, s) ≤ s f (x, s) ⇒ $\frac{f\left(x,s\right)}{F\left(x,s\right)}$ ≥ $\frac{\mu }{s},$ by integration, ln|F(x, s)|≥μ ln|s| +C ‘, hence F(x, s)≥ C|s|^μ.

Proof of Theorem 1.

We will use the above variant of MPT with X = $W_0^{1,p}$ (Ω) and in both cases φ satisfies the condition related to the first assertion. Any such proof contains two steps: 1) The geometry of the MPT and 2) The Palais–Smale condition.

First step. Obviously φ(0) = 0 in both cases (∗) and (∗∗). By usage of Remark 5, ${\int }_{\mathsf{\Omega }}F\left(x,u\left(x\right)\right)dx$ ≥ C${\int }_{\mathsf{\Omega }}|u\left(x\right){|}^{\mu }dx$, hence, for (∗),

$$\begin{gathered} \mathsf{\phi }(tu)=\frac{1}{p}\Vert tu{\Vert }_{1,p}^p-{\int }_{\mathsf{\Omega }}F\left(x,tu\left(x\right)\right)dx=\frac{t^p}{p}\Vert u{\Vert }_{1,p}^p-{\int }_{\mathsf{\Omega }}F\left(x,tu\left(x\right)\right)dx \le \\ \frac{t^p}{p}|\left|u\right|{|}_{1,p}^p-{\mathrm{Ct}}^{\mathsf{\mu }}{\int }_{\mathsf{\Omega }}|u\left(x\right){|}^{\mu }dx, \end{gathered}$$

and, for (∗∗),

$$\begin{gathered} \mathsf{\phi }(tu)=\frac{1}{p}{▍\mathrm{tu}▍}_{1,p}^p-{\int }_{\mathsf{\Omega }}F\left(x,tu\left(x\right)\right)dx=\frac{t^p}{p}{▍u▍}_{1,p}^p-{\int }_{\mathsf{\Omega }}F\left(x,tu\left(x\right)\right)dx\le \\ \frac{t^p}{p}{▍u▍}_{1,p}^p-C{t}^{\mathsf{\mu }}{\int }_{\mathsf{\Omega }}|u\left(x\right){|}^{\mu }dx, \end{gathered}$$

so, when p < μ, for both cases,

$$\underset{t\to +\infty }{\lim }\phi \left(tu\right)=-\infty$$

In order to finish the proof of the geometric condition, it is sufficient to prove that 0 is a local minimum of φ. As |F(x, t)| = $\frac{c}{p}$|t|^p + tα(x) (see Remark 4), $\left|{\int }_{\mathsf{\Omega }}F\left(x,u\left(x\right)\right)dx\right|$ ≤ ${\int }_{\mathsf{\Omega }}\left|F\left(x,u\left(x\right)\right)\right|dx$, hence, for (∗),

$$\begin{gathered} \mathsf{\phi }(u)=\frac{1}{p}|\left|u\right|{|}_{1,p}^p-{\int }_{\mathsf{\Omega }}F\left(x,u\left(x\right)\right)dx \ge \frac{1}{p}|\left|u\right|{|}_{1,p}^p-\left|{\int }_{\mathsf{\Omega }}F\left(x,u\left(x\right)\right)dx\right| \ge \frac{1}{p}|\left|u\right|{|}_{1,p}^p- \\ {\int }_{\mathsf{\Omega }}\left|F\left(x,u\left(x\right)\right)\right|dx \ge \frac{1}{p}|\left|u\right|{|}_{1,p}^p -\frac{c}{p{\lambda }_1}|\left|u\right|{|}_{1,p}^p{-K\Vert u\Vert }_{1,p}={\Vert u\Vert }_{1,p} \left[\frac{1}{p}\left(1-\frac{c}{{\lambda }_1}\right)|\left|u\right|{|}_{1,\mathrm{p}}^{p-1}-K\right], K \in \mathbf{R}, \end{gathered}$$

and hence the conclusion, considering the properties of φ, because 1 − $\frac{c}{{\lambda }_1}$ > 0. To justify the above inequalities,

$${\int }_{\mathsf{\Omega }}F\left(x,u\left(x\right)\right)dx \le {\int }_{\mathsf{\Omega }}\left(c\frac{|u\left(x\right){|}^p}{p}+\alpha \left(x\right)u\left(x\right)\right)dx=\frac{c}{p} {\Vert i(u)\Vert }_{0,p}^p{\int }_{\mathsf{\Omega }}\alpha \left(x\right)u\left(x\right)dx \le$$

$$\frac{c}{p{\lambda }_1} {\Vert u\Vert }_{1,p}^p+ {\Vert \mathsf{\alpha }\Vert }_{0,{\mathrm{q}}^{\prime }} {\Vert u\Vert }_{\mathrm{q}} \le \frac{c}{p{\lambda }_1} {\Vert u\Vert }_{1,p}^p+K {\Vert u\Vert }_{1,p},$$

where

$${\mathsf{\lambda }}_1=\inf \{ \frac{|\left|u\right|{|}_{1,\mathrm{p}}^p}{|i\left(u\right){|}_{0,p}^p}: u\in W_0^{1,p}\left(\mathsf{\Omega }\right)\backslash \{0\}\} (\mathrm{the} \mathrm{Rayleigh}-\mathrm{Ritz} \mathrm{quotient}).$$

For (∗∗),

$$\begin{gathered} \mathsf{\phi }(u)=\frac{1}{p} {▍u▍}_{1,p}^p-{\int }_{\mathsf{\Omega }}F\left(x,u\left(x\right)\right)dx \ge \frac{1}{p} {▍u▍}_{1,p}^p-\left|{\int }_{\mathsf{\Omega }}F\left(x,u\left(x\right)\right)dx\right|\ge \\ \frac{1}{p} {▍u▍}_{1,p}^p-{\int }_{\mathsf{\Omega }}\left|F\left(x,u\left(x\right)\right)\right|dx\ge \frac{1}{p} {▍u▍}_{1,p}^p-\frac{c}{p{\lambda }_1} {▍u▍}_{1,p}^p-K{▍u▍}_{1, p}= \\ {▍u▍}_{1,p} [\frac{1}{p}\left(1-\frac{c}{{\lambda }_1}\right){▍u▍}_{1,p}^{p-1}– K], K \in \mathbf{R}, \end{gathered}$$

and hence the conclusion because 1 − $\frac{c}{{\lambda }_1}$ > 0.

In order to justify the above inequality, it is necessary to highlight:

$$\begin{gathered} {\int }_{\mathsf{\Omega }}F\left(x,u\left(x\right)\right)dx \le {\int }_{\mathsf{\Omega }}\left(c\frac{|u\left(x\right){|}^p}{p}+\alpha \left(x\right)u\left(x\right)\right)dx =\frac{c}{p} \Vert i(u)|{|}_{0,\mathrm{p}}^p+ \\ {\int }_{\mathsf{\Omega }}\alpha \left(x\right)u\left(x\right)dx \le \frac{c}{p{\lambda }_1} {▍u▍}_{1,p}^p+{\Vert \mathsf{\alpha }\Vert }_{0,{\mathrm{q}}^{\prime }} {\Vert u\Vert }_{\mathrm{q}} \le \frac{c}{p{\lambda }_1} {▍u▍}_{1,p}^p + K{▍u▍}_{1, p}. \end{gathered}$$

Since φ(0) = 0 and, from the above two relations φ(u) > 0 for u ≠ 0, it results that 0 is a local minimum for φ.

Step two. In this case, to prove that (PS)_c holds (with c = 0), we take (u_n)_n≥1 in X = $W_0^{1,p}$(Ω), endowed with || · ||_1,p and ▍ · ▍_1,p for the problems (∗) and (∗∗) respectively satisfying: $\left\{\begin{array}{l}\underset{n\to \infty }{\lim }\mathrm{\phi }\left(u_n\right)=0\\ \underset{n\to \infty }{\lim }\left|\left|{\mathrm{\phi }}_w^{\prime }\left(u_n\right)\right|\right|=0\end{array}\right.$. $\underset{n\to \infty }{\lim }\mathrm{\phi }\left(u_n\right)=0$ ⇒ (φ(u_n))_n≥1 is bounded, hence ∃C > 0 e.g., |φ(u_n)| ≤ C ∀n from a rank on. $\underset{n\to \infty }{\lim }\left|\left|{\mathrm{\phi }}_w^{\prime }\left(u_n\right)\right|\right|=0$ ⇒ ($\left|\left|{\mathrm{\phi }}_w^{\prime }\left(u_n\right)\right|\right|$)_n≥1 is bounded, so ∃M > 0 e.g., $\left|\left|{\mathrm{\phi }}_w^{\prime }\left(u_n\right)\right|\right|$ ≤ M ∀n from a rank on. Remember that the norm on X is either || · ||_1,p or ▍ · ▍_1,p for the problem (∗) and (∗∗) respectively, and that on X * is also in relation to the previous two norms. |〈${\mathrm{\phi }}_w^{\prime }\left(u_n\right)$, u_n〉| ≤ $\left|\left|{\mathrm{\phi }}_w^{\prime }\left(u_n\right)\right|\right|$ ||u_n || ≤ M ||u_n || from a rank on. Hence C + ||u_n|| ≥ φ(u_n) − $\frac{1}{M} \langle {\mathrm{\phi }}_w^{\prime }\left(u_n\right)$, u_n〉, and from this relation, we obtain that the sequence (u_n)_n≥1 is bounded. Evaluate now the last expression in cases (∗) and (∗∗):

$$\begin{gathered} \mathsf{\phi }(u_{\mathrm{n}})-\frac{1}{M} \langle {\mathrm{\phi }}_w^{\prime }\left(u_n\right), u_{\mathrm{n}}\rangle =\frac{1}{p}\Vert u\mathrm{n}{\Vert }_{1,\mathrm{p}}^p-{\int }_{\mathsf{\Omega }}F\left(x,u_n\left(x\right)\right)dx-\frac{1}{M}{\int }_{\mathsf{\Omega }}\left(\left|\nabla u_n{|}^{p-2}\right|\nabla u_n{|}^2\right)dx+ \\ \frac{1}{M}{\int }_{\mathsf{\Omega }}f\left(x,u_n\left(x\right)\right)u_n\left(x\right)dx=\frac{1}{p}\Vert u\mathrm{n}{\Vert }_{1,\mathrm{p}}^p-\frac{1}{M}{\int }_{\mathsf{\Omega }}|\nabla u_n{|}^{p}dx+\left(\frac{\mu }{M}-1\right){\int }_{\mathsf{\Omega }}F\left(x,u_n\left(x\right)\right)dx \ge \\ \frac{1}{p}\Vert u\mathrm{n}{\Vert }_{1,\mathrm{p}}^p-\frac{1}{M}{\int }_{\mathsf{\Omega }}|\nabla u_n{|}^{p}dx \ge \left(\frac{1}{p}-\frac{1}{M}\right)\Vert u\mathrm{n}{\Vert }_{1,\mathrm{p}}^p \end{gathered}$$

since we can take M such that p < M < μ. Similarly:

$$\begin{gathered} \mathsf{\phi }(u_{\mathrm{n}})=\frac{1}{M}\langle {\mathrm{\phi }}_w^{\prime }\left(u_n\right), u_{\mathrm{n}}\rangle =\frac{1}{p} {▍u▍}_{1,p}^p- {\int }_{\mathsf{\Omega }}F\left(x,u_n\left(x\right)\right)dx- \\ \frac{1}{M}{\sum }_{i=1}^N{\int }_{\mathsf{\Omega }}{\left|\frac{\partial u_n}{\partial x_i}\right|}^{p-2}{\left(\frac{\partial u_n}{\partial x_i}\right)}^{2}dx+\frac{1}{M}{\int }_{\mathsf{\Omega }}f\left(x,u_n\left(x\right)\right)u_n\left(x\right)dx \ge \\ \frac{1}{p} {▍u▍}_{1,p}^p- \frac{1}{M}{\sum }_{i=1}^N{\int }_{\mathsf{\Omega }}{\left|\frac{\partial u_n}{\partial x_i}\right|}^{p}dx+\left(\frac{\mu }{M}-1\right){\int }_{\mathsf{\Omega }}F\left(x,u_n\left(x\right)\right)dx \ge \\ \left(\frac{1}{p}-\frac{1}{M}\right) {▍u▍}_{1,p}^p. \end{gathered}$$

Hence, we obtained in both cases: C + ||u_n || ≥ $\left(\frac{1}{p}-\frac{1}{M}\right)$||u_n||^p, which means that the sequence (u_n)_n≥1 is bounded. Since X is complete in both cases, it is sufficient to prove that (u_n)_n≥1 has a Cauchy subsequence. To prepare this last assertion, remark that φ = Ψ − ψ, hence Ψ = φ + ψ, where Ψ(u) = $\frac{1}{p}$ ${\Vert u_n\Vert }_{1,p}^p\mathrm{in}$ case (∗) and Ψ(u) = $\frac{1}{p}$ ${▍u▍}_{1,p}^p\mathrm{in}$ case (∗∗) and Ψ(u) = ${\int }_{\mathsf{\Omega }}F\left(x,u\left(x\right)\right)dx$ and ψ’, ψ’(u) =_{${\int }_{\mathsf{\Omega }}f\left(x,u\left(x\right)\right)u\left(x\right)dx$}, is a compact operator.

$$\begin{gathered} \langle \mathsf{\Psi } ‘(u), h\rangle ={\int }_{\mathsf{\Omega }}|\nabla u{|}^{p-2}\nabla u\cdot \nabla hdx, \forall h\in W_0^{1,p}\left(\mathsf{\Omega }\right) \mathrm{in} \mathrm{case} (\ast ) \mathrm{and} \\ \langle \mathsf{\Psi } ‘(u), h\rangle ={\sum }_{i=1}^N{\int }_{\mathsf{\Omega }}{\left|\frac{\partial u}{\partial x_i}\right|}^{p-2}\frac{\partial u}{\partial x_i}\frac{\partial h}{\partial x_i}dx, \forall h\in W_0^{1,p}\left(\mathsf{\Omega }\right) \mathrm{in} \mathrm{case} (\ast \ast ). \end{gathered}$$

We will use the following inequalities [44]: ∀ξ, ζ ∈ Rⁿ,

|ξ − ζ|^p ≤ A(|ξ|^p−2ξ − |ζ|^p−2ζ)(ξ − ζ), for p ≥ 2

|ξ − ζ|² ≤ A(|ξ|^p−2ξ − |ζ|^p−2ζ)(ξ − ζ)(|ξ| + |ζ|)^2−p, for 1 < p < 2.

Thus, we obtain, in case (∗) and for p ≥ 2:

$$\begin{array}{c}{\Vert u_{\mathrm{m}}-u_{\mathrm{n}} \Vert }_{1,p}^p \le a {|\nabla (u_{\mathrm{m}}-u_{\mathrm{n}}) |}_{L^p\left(\mathsf{\Omega }\right)}^p= a{\int }_{\mathsf{\Omega }}{\left|\nabla u_m-\nabla u_n\right|}^{p}dx \le \\ B{\int }_{\mathsf{\Omega }}\left(\left|\nabla u_m{|}^{p-2}\nabla u_m-\right|\nabla u_n{|}^{p-2}\nabla u_n\right)\left(\nabla u_m-\nabla u_n\right)dx=B{\int }_{\mathsf{\Omega }}|\nabla u_m{|}^{p-2}\nabla u_m\left(\nabla u_m-\nabla u_n\right)\\ dx-B{\int }_{\mathsf{\Omega }}|\nabla u_n{|}^{p-2}\nabla u_n\left(\nabla u_m-\nabla u_n\right)dx=B\langle \mathsf{\Psi } ‘(u_{\mathrm{m}}), u_{\mathrm{m}}-u_{\mathrm{n}}\rangle -B\langle \mathsf{\Psi } ‘(u_{\mathrm{n}}), u_{\mathrm{m}}-u_{\mathrm{n}}\rangle \\ =B\langle \mathsf{\phi }\prime \left(u_{\mathrm{m}}\right), u_{\mathrm{m}}-u_{\mathrm{n}}\rangle -B\langle \mathsf{\phi }\prime \left(u_{\mathrm{n}}\right), u_{\mathrm{m}}-u_{\mathrm{n}}\rangle +B\langle \mathsf{\Psi } ‘\left(u_{\mathrm{m}}\right), u_{\mathrm{m}}-u_{\mathrm{n}}\rangle -B\langle \mathsf{\Psi } ‘\left(u_{\mathrm{n}}\right), u_{\mathrm{m}}-u_{\mathrm{n}}\rangle \le \\ B(\Vert \mathsf{\phi }\prime (u_{\mathrm{m}}) \Vert +\Vert \mathsf{\phi }\prime (u_{\mathrm{n}})\Vert ) {\Vert u_{\mathrm{m}}-u_{\mathrm{n}}\Vert }_{1,p}^p+ B{\int }_{\mathsf{\Omega }}\left[f\left(x,u_m\right)-f\left(x,u_n\right)\right]\left(u_m-u_n\right)dx \le \\ B(\Vert \mathsf{\phi }\prime (u_{\mathrm{m}})\Vert +\Vert \mathrm{\phi }\prime (u_{\mathrm{n}}) \Vert ) \Vert u_{\mathrm{m}}-u_{\mathrm{n}} {\Vert }_{1,p}^p+ B\Vert \mathsf{\Psi } ‘\left(u_{\mathrm{m}}\right)-\mathsf{\Psi } ‘\left(u_{\mathrm{n}}\right) \Vert \Vert u_{\mathrm{m}}-u_{\mathrm{n}} {\Vert }_{1,p}^p,\end{array}$$

hence

$$\Vert u_{\mathrm{m}}-u_{\mathrm{n}} {\Vert }_{1,p}^p \le B(\Vert \mathsf{\phi }\prime (u_{\mathrm{m}}) \Vert +\Vert \mathsf{\phi }\prime (u_{\mathrm{n}}) \Vert +\Vert \mathsf{\psi } ‘(u_{\mathrm{m}})-\mathsf{\psi } ‘(u_{\mathrm{n}}) \Vert ) \Vert u_{\mathrm{m}}-u_{\mathrm{n}} {\Vert }_{1,p}^p$$

Similar proof for the case (∗∗), replacing || · ||_1,p by ▍ · ▍_1,p since

$${▍u▍}_{1,p} ={\left({\sum }_{i=1}^N{\Vert \frac{\partial u}{\partial x_i}\Vert }_{L^p\left(\mathsf{\Omega }\right)}^p\right)}^{\frac{1}{p}}={\left[{\int }_{\mathsf{\Omega }}{\left(\left|\frac{\partial u}{\partial x_1}\right|+\dots +\left|\frac{\partial u}{\partial x_n}\right|\right)}^p\right]}^{\frac{1}{p}}\le C\prime {\left({\int }_{\mathsf{\Omega }}|\nabla u{|}^{p}dx\right)}^{\frac{1}{p}}.$$

Since φ’(u_n) → 0 and ψ’ = N_f (Nemytskii operator) is compact, we can assume, passing (if it is necessary) to a subsequence of (u_n)_n≥1, that it is convergent in X.

For 1 < p < 2, in case (∗), ||u_m − u_n ||$\le$ B ‘ | 〈Ψ ‘(u_m), u_m − u_n〉 − 〈Ψ ‘(u_n), u_m − u_n〉| (${\Vert u_m\Vert }_{1,p}^{2-p}+$ ${\Vert u_n\Vert }_{1,p}^{2-p}$). To prove the above relation, we use the statement: ∀s ∈ (0, +∞) ∃ C_s > 0 a constant e.g.,

(x + y)^s ≤ C_s (x^s + y^s), ∀x, y ∈ (0, +∞)

and take: ${\Vert u_m-u_n\Vert }_{1,p}^2\le$ a ‖ |∇(u_m − u_n)| ${\Vert }_{L^p\left(\mathsf{\Omega }\right)}^{2-p}=$ a${\left({\int }_{\mathsf{\Omega }}{\left|\nabla u_m-\nabla u_n\right|}^{p}dx\right)}^{\frac{2}{p}}$. Continue with:

$$\begin{array}{c}{\int }_{\mathsf{\Omega }}{\left|\nabla u_m-\nabla u_n\right|}^{p}dx={\int }_{\mathsf{\Omega }}{({\left|\nabla u_m-\nabla u_n\right|}^2)}^{\frac{p}{2}}dx \le \\ A^{\frac{\mathrm{p}}{2}}{\int }_{\mathsf{\Omega }}{[(|\nabla u_m{|}^{p-2}\nabla u_m-{\left|\nabla u_n\right|}^{p-2}\nabla u_n)\left(\nabla u_m-\nabla u_n\right){(\left|\nabla u_m\left|+\right|\nabla u_n\right|)}^{2-p}]}^{\frac{p}{2}}dx=\\ A^{\frac{\mathrm{p}}{2}}{\int }_{\mathsf{\Omega }}{[(|\nabla u_m{|}^{p-2}\nabla u_m-{\left|\nabla u_n\right|}^{p-2}\nabla u_n)\left(\nabla u_m-\nabla u_n\right)]}^{\frac{p}{2}}{(\left|\nabla u_m\left|+\right|\nabla u_n\right|)}^{\frac{p\left(2-p\right)}{2}}dx \le \\ A‘\left[{\int }_{\mathsf{\Omega }}(|\nabla u_m{|}^{p-2}\nabla u_m-{\left|\nabla u_n\right|}^{p-2}\nabla u_n\right)\left(\nabla u_m-\nabla u_n\right)dx{]}^{\frac{p}{2}}{[{\int }_{\mathsf{\Omega }}{(\left|\nabla u_m\left|+\right|\nabla u_n\right|)}^{p}dx]}^{\frac{\left(2-p\right)}{2}}\le \\ A‘\left[{\int }_{\mathsf{\Omega }}(|\nabla u_m{|}^{p-2}\nabla u_m-{\left|\nabla u_n\right|}^{p-2}\nabla u_n\right)\left(\nabla u_m-\nabla u_n\right)dx{]}^{\frac{p}{2}}{[C_p\left({\int }_{\mathsf{\Omega }}|\nabla u_m{|}^{p}dx+{\int }_{\mathsf{\Omega }}{\left|\nabla u_n\right|}^{p}dx\right)]}^{\frac{\left(2-p\right)}{2}}\le A_p\left[{\int }_{\mathsf{\Omega }}(|\nabla u_m{|}^{p-2}\nabla u_m-{\left|\nabla u_n\right|}^{p-2}\nabla u_n\right)\left(\nabla u_m-\nabla u_n\right)dx{]}^{\frac{p}{2}} ·\\ \left[{\left({\int }_{\mathsf{\Omega }}|\nabla u_m{|}^{p}dx\right)}^{\frac{\left(2-p\right)}{2}}+{\left({\int }_{\mathsf{\Omega }}{\left|\nabla u_n\right|}^{p}dx\right)}^{\frac{\left(2-p\right)}{2}}\right]=A_p\left[{\int }_{\mathsf{\Omega }}(|\nabla u_m{|}^{p-2}\nabla u_m-{\left|\nabla u_n\right|}^{p-2}\nabla u_n\right.\\ \left(\nabla u_m-\nabla u_n\right)dx{]}^{\frac{p}{2}} (\Vert u_{\mathrm{m}} {\Vert }_{1,\mathrm{p} }^{\frac{\left(2-p\right)p}{2}} ||u_{\mathrm{n}} {\Vert }_{1,\mathrm{p} }^{\frac{\left(2-p\right)p}{2}}, \mathrm{so}\\ {\left({\int }_{\mathsf{\Omega }}{\left|\nabla u_m-\nabla u_n\right|}^{p}dx\right)}^{\frac{2}{p}}\le B‘‘{\int }_{\mathsf{\Omega }}(|\nabla u_m{|}^{p-2}\nabla u_m-{\left|\nabla u_n\right|}^{p-2}\nabla u_n)\left(\nabla u_m-\nabla u_n\right)dx ·\\ {(\Vert u_{\mathrm{m}} {\Vert }_{1,\mathrm{p} }^{\frac{\left(2-p\right)p}{2}}\Vert u_{\mathrm{n}} {\Vert }_{1,\mathrm{p} }^{\frac{\left(2-p\right)p}{2}})}^{\frac{2}{p}}, \mathrm{hence}\\ \Vert u_{\mathrm{m}}-u_{\mathrm{n}} {\Vert }_{1,\mathrm{p}}^2 \le B‘| \langle \mathsf{\Psi } ‘(u_{\mathrm{m}}), u_{\mathrm{m}}-u_{\mathrm{n}}\rangle -\langle \mathsf{\Psi } ‘(u_{\mathrm{n}}), u_{\mathrm{m}}-u_{\mathrm{n}}\rangle | (\Vert u_{\mathrm{m}} {\Vert }_{1,\mathrm{p}}^{2-p}+ \Vert u_{\mathrm{n}} |{\Vert }_{1,\mathrm{p}}^{2-p})\end{array}$$

and then the conclusion as in case p ≥ 2. □

Remark 6.

We gave solutions for the above problems (∗) and (∗∗). We can generalize these two problems in:

$$\left\{\begin{array}{l}-{\mathsf{\Delta }}_{p}u\left(x\right)=f\left(x,u\left(x\right)\right) \mathrm{on} \mathsf{\Omega }\\ u\left(x\right)=g\left(x,u\left(x\right)\right) \mathrm{on} \partial \mathsf{\Omega }\end{array}\right.,$$

and

$$\left\{\begin{array}{l}-{\mathsf{\Delta }}_p^{s}u\left(x\right)=f\left(x,u\left(x\right)\right) \mathrm{on} \mathsf{\Omega }\\ u\left(x\right)=g\left(x,u\left(x\right)\right) \mathrm{on} \partial \mathsf{\Omega }\end{array}\right..$$

We can also solve similar Newmann problems:

$$\left\{\begin{array}{l}-{\mathsf{\Delta }}_{p}u\left(x\right)=f\left(x,u\left(x\right)\right) \mathrm{on} \mathsf{\Omega }\\ \frac{\partial u}{\partial n}=g\left(x,u\left(x\right)\right) \mathrm{on} \partial \mathsf{\Omega }\end{array}\right.,$$

and

$$\left\{\begin{array}{l}-{\mathsf{\Delta }}_p^{s}u\left(x\right)=f\left(x,u\left(x\right)\right) \mathrm{on} \mathsf{\Omega }\\ \frac{\partial u}{\partial n}=g\left(x,u\left(x\right)\right) \mathrm{on} \partial \mathsf{\Omega }\end{array}\right..$$

The assumptions for the Carathéodory function g: ∂ Ω × R → R are: |g(x, s)| ≤ g₀(x) + g₁(x) |s|^p−1, where g_i: ∂Ω → R, i = 0,1, are measurable functions, g₀ $\in L^{\frac{p}{p-1}}$(∂Ω) and 0 ≤ g_i(x) ≤ C_g, ∀x ∈ ∂Ω. Suppose also that there exists μ₁ > p, e.g., μ₁G(x, t) ≤ tH(x, t) a.e. on ∂Ω and ∀t ∈ R, and there exists a nonempty open set U ⊂ ∂Ω with H(x, t) > 0 for (x, t) ∈ U × ∂Ω, where G(x, t) =${\int }_0^{t}g\left(x,s\right)ds$.

Remark 7.

The involvement of the weaker version from [40] of MPT entails the possibility of weakening the conditions in any application.

Remark 8.

Regarding the usage of versions of the mountain pass theorem applied in problems even more general than those mentioned above, one can specify that they can be either indirectly used (like here, via Theorem 1) or directly as in the papers [45] to characterize positive solutions for nonlinear singular superlinear elliptic equations, [46] where a Dirichlet problem for resonant anisotropic (p, q)-equations has been solved or in [47] in which positive solutions for parametric (p(z), q(z))-equations are described. In all those cited results, one of the key steps is based on a MPT variant.

3. Discussion

In many mathematical analysis problems, the aim is the construction of a suitable solution. “To construct” often means to obtain a solution in a “weak” sense not only since they are taken in a “distributional” or Sobolev sense, but because the existence proof of the solution is often not constructive (even though one may often additionally find some qualitative or quantitative properties of these solutions).

Sometimes, the problems considered exhibit a variational structure, i.e., the desired solutions can be obtained as critical points of a functional (in the literature, this function is called “energy”, though, from a physical point of view, this is more related to a “Lagrangian action”).

When the problem is of variational type, the approach can be made using any me-thods aiming to prove that a critical point actually exists for the function involved. Some of these methods are like “natural” generalizations from calculus toward advanced functional analysis, where the solutions can be found corresponding to local (or sometimes global) minima of the functional.

We are interested in proving the existence of entire weak nontrivial solutions for some classes of nonlinear partial differential equations. The differential operator is described by potential with different growth near zero and at infinity. The proofs combine arguments based on the mountain pass and energy estimations.

Starting from a namely variant of the mountain pass theorem, the main existence result for the solution of two Dirichlet problems with p-Laplacian and p-pseudo-Laplacian is proven. For this proof, two specific norms on $W_0^{1,p}$(Ω) have been introduced, and certain goal functions are used. The demonstration consists in showing that those function-nals, under the conditions from the central result, fulfill the conditions from that variant of MPT. The importance of this type of approach is given, among others, by the construction of the solution via a namely numerical method. For our future research development, we are interested in the usage of Computational Fluid Dynamics to draw and visualize the required solutions. In the following, some important applications in models for namely real phenomena are presented.

3.1. Applications in Modelling of Real Phenomena

3.1.1. Nonlinear Elastic Membrane

I.

To characterize a nonlinear elastic membrane under the load f, we can use the following mathematical model:

− Δ_p u = f on Ω,

u = 0 on ∂Ω.

The solution u stands for the deformation of the membrane from the rest position [32,48].

The nonlinearity of the p-Laplacian is often used to reflect the impact of nonideal material to the usual vibrating homogeneous elastic membrane modeled by the Laplace operator. In this case, the deformation energy is given by ${\int }_{\mathsf{\Omega }}|\nabla u{|}^{p}dx$. Therefore, the greatest lower bound on $W_0^{1,p}$(Ω)\{0} of the Rayleigh quotient is usually referred as the principal frequency of the vibrating nonlinear elastic membrane. Consider the problem:

− Δ_p u = |u|^p−2 u, x ∈ Ω,

u|∂Ω = 0.

This can be considered as (∗) with f ( ·_, u( · )) = |u|^p−2 u. f (having this form) fulfills the two conditions—properties of the Carathéodory function taking c = 1 and α ≡ 0 in order to fulfill the growth condition. Hence the last problem has a solution on $W_0^{1,p}$(Ω) in the sense of W ^{− 1, p’} (Ω).

II.

In [49] is proposed a model for the vibration of a nonhomogeneous membrane which is fixed along the boundary. Several materials (with different densities) are investigated following the location of these materials inside Ω by studying the first mode in the vibration of the membrane. Ω is a bounded smooth domain in R^N, g is a Lebesgue measurable function verifying the condition 0 ≤ g(x) ≤ H, ∀x ∈ Ω, where H is a positive constant, g$\cancel{\equiv }$0, H. g can be replaced by any Lebesgue measurable function equal to it almost everywhere. Consider the eigenvalue Dirichlet problem:

− Δ_p u = g(x) |u|^p−1 u, x ∈ Ω,

u|∂Ω = 0.

To superpose our theory, it is sufficient to take f (x, s) = g(x) |s|^p−1, so | f (x, s)| = |g(x)| · |s|^p−1 ≤ H|s|^p−1, hence the growth condition is fulfilled with c = H, and, by applying the above Theorem 1 (its conditions being fulfilled), the last above problem has a solution in $W_0^{1,p}$(Ω) in the sense of W ^{− 1, p’} (Ω).

III.

Using Theorem 1 from the previous section, we can propose a proof for the existence of the solution of the nonlinear problem of the elastic membrane under the load f, in the general case when f is a Carathéodory function that fulfills the growth conditions, and the others from the cited result.

3.1.2. Application in Glaciology

Consider the elliptic partial differential equations for which the principal part Δ_p u is applied in physics, for instance, in the description of phenomena in glaciology as the sli-ding of glaciers, see [2,50]. In [51], a Dirichlet problem is formulated:

− Δ_p u = f on Ω,

u = φ on ∂Ω,

with p = 1+$\frac{1}{g}$, g being the Glen exponent, where 0 < g < +∞, hence 0 < p < +∞ and, for a real glacier, g ≥ 3, and usually taken, g = 3, so p =^.$\frac{4}{3}$ and f satisfies the necessary conditions from the statement of Theorem 1. Replacing u by v = u − φ on the boundary of Ω in the sense of the trace and u ≡ v on the interior of Ω, one can apply Theorem 1 from Section 2 to give another proof for the existence of the solution for such a problem.

3.1.3. Nonlinear Elastic Membrane with p-Pseudo-Laplacian

• Let us take the expression proposed in [52]: ${\int }_{\mathsf{\Omega }}{\sum }_{i=1}^n{\left|\frac{\partial u}{\partial x_i}\right|}^{p}dx$ for the energy of deformation of the membrane when it is woven out of elastic strings in a rectangular form. Then the phenomenon can be modeled with a Dirichlet problem for the p-pseudo-Laplacian:

$$-{\Delta }_{\mathrm{p}}^{\mathrm{s}}u={\left|u\right|}^{p-2} u, x\in \mathsf{\Omega },$$

u|∂Ω = 0

One can apply Theorem 1, when p > 1, taking there: f ( ·, u( · )) = |u|^p−2 u, with c = 1 and α ≡ 0 to fulfill growth condition. Show that the rest of the conditions from the main result are fulfilled, and then we can obtain that the last exposed problem has a solution in W₀^1,p (Ω) in the sense of W ^{− 1, p’} (Ω).

II.

Take in the above problem I the load f in the general case when f is a Carathéodory function which fulfills the conditions of Theorem 1 and hence we can generalize I quickly.

3.2. Prospects and Future Developments

Important modeling tools for processes associated with anomalous diffusion or spatial heterogeneity are the fractional differential equations being increasingly used [53]. Power law diffusion equations involving p-Laplacian with constant and/or variable p present considerable importance in connection with studies on non-Newtonian fluids, turbulence modeling phase transitions, data clustering, machine learning, and image processing. A lot of studies related to power law diffusions need the development of efficient numerical methods for solving elliptic partial differential equations with p-type gradient nonlinearities [19]. The fractional models have been used to describe chemical or conta-minant transport in heterogeneous aquifers in water resources. In the context of magnetic resonance, fractional models of the Bloch–Torrey equation describing anomalous diffusion are considered. Anomalous diffusion in cell biology has been measured in fluorescence photo-bleaching recovery, and for simple types of chemical reaction-diffusion equations, to describe microscale diffusion in the cell wall of plants, fractional-in-time models are developed. In [54], Liu and Burrage have fitted parameters to time fraction gene re-gulatory models for a bacterium found in soil, Bacillus Subtilis. Such a problem is important from the theoretical point of view, and also appears in many physical applications and describes a vast pallet of models in applied sciences. This is used in the models of chemical reactions, heat transfer, population dynamics, and others [12].

Of the highest interest will be the design of such mathematical models for simulation of surface phenomena involved in micro-emulsification processes [55,56,57] applied in nanofabrication, where the role of the chemical reactor in nano-liquid–liquid dispersed systems is played now by micro- or nano-droplets, and the determinant parameters are special intermolecular forces at interface. In this context, new research directions are opened to extend the applicative pallet of this way to construct the solutions needed for modeling real processes that will be developed within new emerging technologies.

4. Conclusions

An MPT variant has been involved in proving existence results for two mathematical physics problems with p-Laplacian and p-pseudo-Laplacian. The novelty consists here in the application of this type of MPT result in the proof of the main result and also the usage of the main theorem in solving problems with these kinds of operators.

The implication of the main result in drawing the solutions for some important mathematical physics problems opens a door toward the solutions of other parallel models of real-world phenomena.

Our interest is now focused on such kinds of problems and their solution, and this is an introductory study for future developments to find out a mathematical model (there is no one available) to be applied for describing diffusion phenomena involved in micro-emulsification of disperse systems in tight connection with surface properties at the interface in self-organized systems. These mathematical physics approaches are at the basis of nanofabrication—one of the most promising trends within new emerging technologies.